Welcome to Veigar. I'm not going to call this the end-all guide to playing him. I've seen successful Veigars that do not farm as much. However, there is so much potential lost. Some things out of the ordinary in this build. Most notable is Lich Bane instead of Deathfire Grasp, which will be explained later.
Enjoy and comment. I'll try to get on and make changes if people have a big problem. Though I'm unlikely to change how I do it, and thus unlikely to change much here. I can definitely add more gameplay info if people run into major problems. And I apologize, I know Vlad is going to eat your face mid, he's a really hard mid to counter and you will have to deal with it.
Don't ditch it until you try it, and it is a hard gameplay if your enemy mid knows what they are doing.

Also, the AP in game will be much higher than stated, due to farming (and it looks like archangels isn't taken in to effect. Dunno about that.) Usually in a 50 minute game with this build I end around 1000-1050 AP.

Please note, this build is how I do it. If you violently disagree with something, then change it. If the game is going really differently, then be situational. Luckily, Veigar doesn't have to be situational very often. Either you win, or your team can't hold and they eventually push you over. I'm not going to get in to describing alternatives to items. These are what work the best for me, you know what you need to counter whatever is killing you, and you can do it. You don't need me to tell you how to adapt, either you can or you need to learn. I'm here to tell you how to function as Veigar, the Tiny Master of Evil.

Pros:
Best burst in the game, simply because of infinite AP build and cast-and-watch approach. (As opposed to Malzahar or Leblanc or Annie that have to stay around to finish the job.)
Powerful to the point of hilarity endgame if your farming goes well.
Stupid huge damage, particularly to AP champs.
Best anti-carry in the game, giving your carry free reign to kill their team.
Forced to farm, thus huge CS and lots of gold.
Really, really OP stun. Just.. yea.
Crazy splash damage if you can catch multiples with dark matter.
Virtually undrainable mana pool lategame.

Cons:
Hard to lane, especially vs certain champs (not always the regularly difficult mids)
NEEDS MID ALWAYS
Usually survivable, but stun is highly vulnerable to certain champs.
Focused first
Rough cooldown on stun, particularly early on.
Useless after first burst, though this is resolved partially with lichbane and fast Q's.
You'll want to gouge your eyes out the eight hundredth time you hear "STOP LAUGHING, I AM EVIL!"

I use magic penetration marks (pretty much the best marks out there) flat mana regen seals, and cooldown/level glyphs. (Veigar doesnt need mana regen late game, his passive and the archangel scepter takes care of that. Early on, however, the mp5 seals+meki pendant+his passive let him farm without pause. The cooldown glyphs make up for lack of CDR in the item build itself, and come out to around 17% CDR with masteries. HP quintessences give a little resilience, though some move speed quints would also be useful.
Some others that seem like okay ideas:
Probably the first to be substituted out would be the HP quints for MS quints, maybe some mpen as well.
The Mp5 seals could go, not having them may slow down your early game, but at the same time you could have some bonus AP or movespeed, for that brand that would otherwise totally destroy you.

9/0/21 Standard caster masteries. I like the CDR on flash and teleport. I suppose you could go with a 21/-/- for more damage, but its not necessary, your burst is enough. Maybe -/21/- for more survivability as well.

Start out with a meki pendant and hp pots, and stay in your lane farming as long as you possibly can. when you go back, upgrade to a tear and buy a catalyst and boots if you can afford them all. Get ruby first if you can't afford the whole catalyst.
Sometimes it is necessary to grab an early pair of boots, i.e. Brand and Karthus.
When you finish your sorcerers shoes farm up for blasting wands. I do both before upgrading because since the RoA and archangels supplement each other, its better to buy them simultaneously and stock up on straight AP for some extra punch until you can buy both, and your tear will be farming on its own already. Upgrade to Archangels Scepter and Rod of Ages next, they supplement each other well and at this point you will have a huge mana pool and around 350 AP if you farmed well.
Rabaddon's next, simply because.. duh. It's Rabaddon's. It will kick your AP from ~350-400 before to ~650-700 after.
I do not like Deathfire Grasp on Veigar, I will get to that later when I talk about Lich Bane.
Push on to a Void Staff to make your burst unsurvivable.
I do not stack archangel's scepters. I used to, but the extra ~500AP (1500 total) isn't worth the damage lost to even the smallest magic resist, and the damage lost to not having Lich Bane.
Lich Bane is last. Here come the comment arguments. (They are welcome.)
I choose Lich Bane over DFG because DFG is capable of a (maybe) 1500 dmg burst once in a fight, and thats on a tank (who you shouldn't be aiming) at full life. Lich bane, along with your ~1000 AP deals around 850 damage after resists EVERY TIME YOU CAST YOUR Q. So, in a 12 second fight, thats four thousand damage from lich bane as opposed to maybe one thousand from DFG. Also, lich bane is a great finish to your burst if the enemy counter with banshees, (which is likely) and you should always break banshees with Q, because stun and W deal far more damage than Q.

Baleful Strike first, and always. Tips say to start harassing around rank 3, but its not really worth it. Only harass if you get harassed. Or if its a crazily easy kill on a champ that never builds hp. (Karthus is a good example.) Farm always, AT NO POINT IN THE GAME SHOULD YOU NOT FARM BALEFUL STRIKE. EVER.

Grab your stun at level 2, to ward off ganks and save your *** many, many times. Also works if they tower dive.

Note with stun, some champions can dive through it and come out okay. Most notable is Xin, who can dive through your stun, land on you, and pop you up before you can make it out of the other side to try to stun him again. This sucks. Others include Kassadin, LeBlanc, Ezreal, anyone with Flash or flash-like abilities.

Dark matter at level 4, but ONLY use it if you are getting multiple waves pushed to your tower, or have a guaranteed kill. It hurts your mana too much to use, and takes kills your Q could have.

Primordial Burst at level 6, duh. Note about this move. People call Veigar an AP killer only. Don't just kill AP champs. Yes, it does great damage to AP champs, but its not that it does less than normal to non-APs. Think of it as doing bonus to APs, and still epic damage to the non casters. Veigar has the burst to bring down anything but high MR tanks endgame, don't reserve your burst for a poorly played AP when you could bring down the carry that is killing your team.

Continue R>Q>W>E. I level dark matter before stun because level one stun is enough (as long as they don't have tenacity) to land dark matter, and dark matter levels make your burst better, as well as the ability to drop caster waves at level 10 (rank 3) assuming your farming is doing well enough. Note, once your W is able to burst caster waves, it is also able to take down the wraiths (assuming a jungler hasn't been beefing them up.) Take the wraiths whenever you are near and have mana/CD on W.

I take Teleport and Flash. Flash saves you, and throws you in range for your OP stun. I like teleport for versatility and lane sustainability, (I always feel helpless with teleport.) Veigar is great for Teleport because he is so good at stopping pushes. Also takes care of that pesky backdoor Master Yi/Sivir.
Also solid choices are ghost and ignite. Ignite is great early game for the burst help, though endgame your burst won't need ignite to kill everyone you see. Endgame, use it on the lifetap/spellvamp carry you cant burst. (Akali, Vlad, any AD carry, particularly Xin.)
Acceptable is exhaust, I guess, but if you are in range to exhaust you are in range to kill outright, and exhaust makes it so they do less damage, but Veigar's whole point is to kill before they get that chance. Ghost is better for chasing/running.
Pretty much everything else is a bad idea, even though Heal isn't terrible in my eyes. It can win you those early burst fights with an Annie and then you snowball and their day is fairly ruined by that point.
Don't use flash if you can help it, you almost always need it to save your ***. Unless you can make it to your tower as soon as you kill your lane enemy. Be very wary of pushing/ganks when flash is down. Stun will help, but not against some, particularly Xin.

Veigar's play is very easily explained, but takes a inversely large amount of practice to make solid.
ALWAYS TAKE MID. JUST.... ALWAYS. TAKE. MID. Veigar is pushed too easily in lanes, and loses last hits to allies. Top solo is acceptable only if the enemy team is ALSO top solo, as two champions will have your tower down in under five minutes.
Start with Baleful strike, use it always. Last hit with Q, or auto attack if Q is on CD. First 6 minions should be all yours. You won't have level 3 when the cannon minions come, so be wary trying to last hit against them because they often will kill before your Q arrives. It is best to use an auto attack followed by Q to ensure the kill. If you think Q will kill, then and auto+Q will always kill. At rank 2 of Q, you can one shot casters after the tower hits them once. Starting at rank 2-3 of Q, autoattack a melee minion at 1/4 life followed immediately by Q to ensure last hit, maybe higher if it is the focus of your minions/a cannon minion. Range of Q is equal to your attack, so by attacking you will always be in range to Q, unlike Annie, for example.
DO NOT HARASS THE ENEMY UNLESS THEY HARASS YOU.
Veigar is easily pushed, so focus on autoattacking the enemy creeps down as well as using Q. If a confrontation does occur, autoattack the enemy only away from their creeps. If an AD champ is mid, let them try to hit you once and just hit them back. They will end up taking more damage via your minions than they deal to you, if you are not overextended. Use healing pots when necessary, and stay and farm AS LONG AS POSSIBLE. this means to the point (and past the point) of casting Q EVERY TIME you get 75 mana, because you are using it so much to last hit. Don't try to burst the enemy out at level 6, you won't have the mana for all four spells. Also, suggest your jungler gank elsewhere because it throws off Veigar's farm. Disregard this if the enemy is a hard mid (Vlad and Brand in particular.)

If you find yourself being zoned, skirt around and land whatever Q's you can. Also, realize that your turret can be one of the best weapons you possess, especially for your farm.

Keep in the laning phase AS LONG AS YOU CAN. The more you can farm the easier your bursts will be. Nothing hurts Veigar more than when the "early game" phase ends too soon. Try to keep teamfights out of your lane. However, if there is an easy kill somewhere, do take it. Exchanging kills is not terrible for Veigar, as you get more from a kill (AP) than they do.

Farm, farm, farm. Continue to burst and snowball your opponent out of the lane if you can. Once you go back and get tear and catalyst, you will be able to use your mana much more generously. Throwing a Q at the enemy champ every 3-4 Q's on minions will be enough to either zone them out, or kill them if they try to stay.

Rinse and repeat.

One laning phase ends, take any kill you can get all the time. Its 5 AP every one you get. Also, NEVER STOP Q-ING MINIONS. EVER. You will hear that at least one more time before this is over.

Burst technique:
If you can get a Q in immediately pre-burst, then do it. Then Stun, Dark Matter, Primordial Burst, Baleful Strike.

Sooo looks like this in button terms. Q-E-W-R-Q OR just E-W-Q-R if you can't get in range to lead with Q. The reason to Q before ult is because your q has a slightly lower range than your ultimate.

By the time you have AS and RoA this burst should kill the squishier champs. Upon Rabaddon's acquirement, you will have the AP to burst pretty much anyone apart from the tank.

If you find yourself getting ganked, or someone will kill you and your stun is down, you would be surprised at how many people will chase Veigar even though it means getting nailed by dark matter while running. Maybe they think it won't actually hit them, or do much damage, but you can get a ton of kills by firing W a few steps in front of you and running through it, with some variation depending on how close they are behind you. It takes aim, and a little practice, but it's pretty epic taking down a full hp master yi without your stun when you have 10% hp.

TEAM FIGHTS
If the fight engages quickly, do this:
Throw a stun and aim for as many as you can. Then drop a dark matter on whoever is most dangerous/squishiest and follow with Q+R (unless your team is also hitting them, in which case the R may not be necessary. If you can save R then you will have your full burst back in the 10 seconds that it takes stun to cooldown.) Ideal is saving R, but don't let a kill get away because you assumed you would get them without it.
If the fight engages slowly, lots of poking, do this:
Range on stun is really, really huge. Throw it out and burst out the poker, because they are often squishy. (Cait. LOL. or Corki. Some others) This way the fight will start 5v4 with your team on the advantage, and your cooldowns will be in a better condition.
Again, don't worry about only focusing the AP. Endgame, you will overkill even the AD carries by 500-1000 damage, so it doesn't really matter.
AVOID XIN ZHAO. He does unspeakable things to Veigar because of his stun-diving and long attack range, combined with knock-up. Keep him on the far side of your stun, out of range of targeting you. (AKA don't put the stun around yourself as a shield, it won't save you from Xin.)
Avoid people with high magic resist, but if that Force of Nature Mordekaiser is totally fisting your team, then yea, take him out. It may not kill him but it'll do enough that your team will bring him down without much difficulty.

You can make or break the game for your team, but remember that Veigar is not a hard carry. He can take out 2 or 3 if you are lucky, but he will never be the carry than Akali, Katarina, or any of the AD carries can be. His cooldowns are too high and he is not sustainable enough.
Play him well, and you will have the enemy team RAGING, and it'll get worse, especially after the third or so time you knock out both their carry and their support in one burst.
Just don't let it go to your head. Veigar is no Rambo.
Oh, and I said you would hear this again, soooo....
NEVER STOP FARMING WITH Q. EVER. IT IS NEVER TOO LATE.
Good luck.

I'm gonna go by what pops into my head as really annoying.
In general-If the mid isn't being aggressive, don't take it as a change to pressure them, as they may change their play. If they are passive, take an advantage and farm as long as you can, you will kill them harder later. This applies to champions that are notoriously hard to deal with, but you don't have to worry. Let them push, and just farm. Kill them later. (Nunu, Morde)
Jax-
Jax hurts, because its hard for Veigar to counter harass and keep up his farm, and Jax has a harsh burst with no ally to ward him off.
Vlad-
There is no way to counter harass Vlad, too much healing, and his burst is a bit rough at six. Start harassing around level 7 and you should be able to burst him out.
Annie-
Be wary of Annie's stun. Veigar is a passive champ so it isn't too difficult to keep her out of range. However, her burst IS better than yours at level 6, and her stun is easier so she will usually win the first burst fight. Try to avoid it.
Ashe-
Stand behind minions, dodge cone, you win. Watch out for her engaging with her ultimate at 6, it usually happens this way against Veigar.
Morde-
Not as hard as with many others, because the goal isn't to kill him. Once you are 7-ish, start a hard pressure and go for the kill, or better yet, a gank on your lane.
Vayne-
This can be rough, if she is super aggressive. Deal with it like Ashe, she has short range so you can fight back and cause your minions to do some real damage to her if its really early. After that, you win the bursts.
Akali-
Her thing is to drop a bubble and punish enemies that don't back off. You have no reason to be on top of her, so don't be.
Ryze-
Easy, your burst is better, just watch his Q harass, it's quite fast.
Malzahar-
Really easy to farm against, just don't get randomly caught by visions. At level 5, feel free to exchange Q for his visions on you, you'll win the burst because you can far outrange him with your stun and kill him before he has a chance to do anything.
Caitlyn-
Ugh. Watch out for peacemaker, also stay back because she has stupid, stupid range on her auto. Run strangely and last hit.
Miss Fortune-
Not as bad as Caitlyn, just watch out for double up. Its fairly frightening in the earliest stages of the game.
Brand-
Yeaaaa. Brand is a hugely OP mid. If you expect Brand, don't buy anything and check. If he is mid, go back and buy boots and three health potions, then buy meki asap. Or wait until you can afford all of the tear. If you can get a really early gank, you will win. With Brand it is entirely an issue of who can start snowballing over the other first.
Nunu-
Booo. Stay out of range, don't try to kill because you will only waste mana. Kill once you have fairly heavy output from an easy farm.
Weird mids- (Garen, etc.)
The tanks actually do surprisingly well against Veigar because he has to farm. But just farm around their annoyances.
Kog'Maw-
Dont let him harass you too hard. Stay out of range of slow and move unpredicatbly. One slow can be quite a lot of your HP in auto-attacks. Burst when ready, and kill.
Sivir-
Squish squish squish, this one isn't too bad. She'll farm, you'll farm, oh well.
Nasus-
Neither of you can prevent the other from farming. His q gets big, so does yours.
Urgot-
He can be really, really tough. Dodge the canister and you'll be fine, if not, he'll totally mess up your day.
Morgana-
Usually too tanky to kill outright, and shield makes your stun (and thus half your burst) useles. Just farm.
Twitch-
Eaaa***yyyy. He will never be able to harass you more than you can harass him, and then he is a crazily easy squishy kill.
Ezreal-
Easy, stay behind minions, hard pressure at level 7, he runs, you win.
Kennen-
Same as above with Ez. Watch how he handles his W's passive mark on his autos though.
Orianna-
I've never laned against her mid as Veigar, but I play her mid all the time. I suggest to watch the ball, and move really really really curiously.
Leblanc-
Watch the silence, other than that you are okay. Her passive is useless because you burst to kill, and as long as you can kill the meteor hits last and thus does splash while she is invisible, still killing her. Just make sure you get your nukes off before she splits.
Zilean-
This can be rough. Sit back and farm and hard pressure at around 7, ganks are best for him though.
Karthus-
Easy. Dodge around his lay waste, and you win. Karthus rarely builds tanky and he can't stop you from killing him. Easy harass and kill by level 6-7. His wall only helps him kill you, it doesn't stop you, and since you do all your burst straight up, you will usually win. Just don't be close enough for his AoE melt. If you are, then you deserve to die.

Can't think of any others now. Generally, if they have overextended at low hp and you can take the kill, do it. Or even do it if you can flash to them and take it. But DO NOT expend all your effort towards the kill. If there is an opening jump on it, but don't waste your farming resources trying to kill someone. Doing so will hurt you in the end.

Keep an eye out for junglers and MIA's from lanes. However, with flash and stun, you will rarely get successfully ganked. Also, because Veigar is pushed easily and pushes not so easily, he is a hard gank target.