This is my first guide, so go easy on me.

I'm making this guide because Veigar is by far my favourite and best character.
I love Veigar so much because he is 100% troll. You just have to turn up the volume, look at the way he runs or do his dance in game just to see this for yourself.

Veigar is one of the few champs in game that is an excellant anti-ap nuke purely because of his ulti which hits 80% of his targets ap (+a certain amount). This means he kills certain champs with ease even if he is not fed (such as Xerath).

He is undoubtedly a mid champ and should be played as such.

The only problem with him is that if you miss your stun you are most probably dead.

Pros (good points):

Veigar nukes squishy ap nukes and ad carrys with ease.
Characters that are shorter range than him are useless in lane. (Like Annie.)
His Q stacks ap on kills, meaning if you can last hit you will be a good Veigar.
Due to this he is one of the best characters to use Deathfire Grasp well.
He uses no real skillshots and so is easy to get used to.

Cons (bad points):

He is single target, meaning if you use your nuke on a tank, you will probably lose the teamfight.
If you buy a Deathfire Grasp as your first item he is incredibly mana hungry and you should probably aim not to harass.
His Dark Matter (W) is almost useless unless you get ganks, as it takes last hits and has to be combined with his Event Horizon (E) to hit competent targets.
He works really badly in lane against long range characters ( Xerath, Galio, Etc.)

Greater Marks of Insight - Magic Penetration: I use these because every character in game has base Magic resistance, you may as well chip away at the percentage reduction it causes.

Greater Seals of Clarity/ Replenishment - Mana Regen: You will need insane mana regen on Veigar to make his Baleful Strike (Q) last hits work well as at level 3 and above they become very mana hungry. This can be countered by mana regen runes, masteries or an early Catalyst the Protector.

Greater Glyphs of Potency - Ability Power: NUKE. 'nuff said.

Greater Quintessences of Potency - Ability Power: NNNNNNUUUUUUUKKKKKKEEEEEE!!!!

With Veigar the aim is to damage your opponent as much as possible in the smallest amount of time. This is why you should pretty much get every single aggressive AP mastery there is.

The mana and mana regen masteries are there for last hitting with your Baleful Strike (Q), BECAUSE YOU MUST DO THIS.

The extra magic resist was just leftover, and since you're in mid and most likely to be against AP you might as well.

Sorcerer's Shoes: Great Magic Penetration - Allows you to hit more, Don't let anyone tell you that lucidity are better, they are not.
Rod of Ages: Mana Boost - The mana from the passive is ssssssooooo nice on Veigar and the extra AP, and health allows you to nuke a bit earlier and still stay in lane.
Deathfire Grasp: NUKE - If you can use this in conjuction with his stun then they drop quickly, from a very early level.
Rabadon's Deathcap: More AP - Need I say more?
Hextech Revolver: You will likely die in most teamfights. But only if you don't have your spell vamp.
Rylai's Cystal Sceptre: Help your teammates to catch that Soraka that always runs from teamfights if you don't have your stun up.
Lich Bane: - A bit more omph: Without using any abilities you can hit another 700 odd damage late game with a Lich Bane.

Other Items:
Void Staff/ Abyssal Sceptre: A bit more magic penetration, perhaps overkill as your targets should be AP or AD carrys, with little mage resist.
Doran's Ring: A good starting item, however it makes you easy to poke out of lane if you have a lack of health pots.
Mejai's Soulstealer: I wouldn't. It's to easy to die and you probably won't get many kills early game making this item pointless.

Level Your Baleful Strike (Q) as it is the ability that stacks your AP, making it easier to last hit.

Your Ulti must always be levelled at 6, 11 and 16. It just should.

Dark Matter (W) does not need to be upgraded very strongly because it is to tempting to use and you need to conserve most mana for last hitting with your Q. However you should upgrade it slightly for some more damage in ganks.

Event Horizon (E) needs to be about level 3 to get all three of your other abilities off and get out before they can do anything. The stun is extremly helpful in teamfights.

You need to farm well to use Veigar. If you have your Baleful Strike (Q) on cooldown then last hit with auto attacks. You do not want to push the lane towards the enemy turret and so you should only last hit the minions.
If you run out of mana then go back to base, asking the jungler to hold your lane. You should not waste money on mana pots as they equate to about 2 lasts hits... if you hit them...
Blue buff is extremely helpful to Veigar as he can concentrate completely on last hitting, taking his stupid mana usage out of the equation.

You should aim by the eng of the game to have collected at least 500 ap by the end of the game.
Keep practising last hits, although Veigar needs them as a character, he has not got a very good minion farm.

Generally have fun playing Veigar, he is a good character to make you feel better about yourself due to the help your stun can provide to your team.

Oh, and look out for enemy junglers, usually they are useless against your stun, however some aren't and it catches you out. Maokai, Blitzcrank (I know this sounds stupid) and Master Yi are good examples of this.