Veigar is one of the best heroes in LoL no doubt. He has great ap-scaling, extra ap from abilities, a reliable instant area of effect (aoe) stun and an ultimate that makes it essential for all enemy ability power users to stack magic resistance. I can never express fully how good he is.

I will now tell you how I play veigar; not with items that leaves him vulnerable to cc or where an ambush means certain death. You will be able to move very freely while using this build and not be as teamreliant as most glass cannon builds are. I'm not saying my build is better, but I personally can't play a glass cannon build and therefore has made this build which I find is incredible good and gives me good results in all games.

17/2 - 2011: Some minor edits made as I noticed some new things while climbing the elo ladder and because of patches etc. For example, DFG is viable lategame, I use this now.

1/3 - 2011: Big big changes made. I get 2x Philosopher stones now at start instead of catalyst which changes my entire itembuild (thanks Countic). The spell progression has been changed as well.
Playstyle is pretty much the same though. Also here is a replay of one of my games showing my playstyle (needs LOLReplay to play), enjoy!

Greater Mark of Insight:
Magic resistance penetration is something all ap-users need. It is very hard to get ingame, and is also very expensive. Works as good early game as at the end. It is a solid choice. No optional runes here.
Greater Seal of Clarity:
Mana regeneration runes are quite necessary for veigar, as he is very mana thirsty throughout the whole game. No optional runes here either.
Greater Glyph of Focus:
Flat cooldown reduction runes work very good with Veigar because of ap-farming with Baleful Strike (more on that later). Optional rune choice is Greater Glyph of Force for more ability power (although 27 more ap at lvl 18 isn't going to make much impact on gameplay).
Greater Quintessence of Fortitude:
Veigar is a squishy champion at start, this is a solid choice and it'll serve you well throughout the whole game. Optional runes are Greater Quintessence of Desolation (for more magic resistance penetration) or Greater Quintessence of Potency (for more start ability power and therefore easier last hitting with Baleful Strike).

It's a standard Ability power-users build giving you:
Cooldown reduction on abilities
Cooldown reduction on summoner spells as well as an upgrade to ghosts' speed
Mana regeneration
Magic resistance penetration
A little extra AP
And a little movementspeed improvement

At start:
Buy two Faery Charms and 3 health potions which will build into 2 Philosopher's stones. Pick Baleful Strike as your first spell and head down whichever lane you want. Veigar himself does poorly until he can buy the stones in any lane and therefore lane choice is mostly irrelevant. I used to take whatever I got, but with the incredible health and mana regeneration from the stones, you are a solid solo laner and I prefer any solo lane.

Early game:
Start last hitting all you can with autoattacks and preferably Baleful Strike without getting harrased much. Focus on this and this only, ganks etc are for other people. You need to farm. Hopefully, by the time you need to go back and heal you will at least have enough gold for the Philosopher's stones (1240) but chances are you dont. Just get one and if possible the Boots of Speed. Continue to farm with ease and dont rush it, the only way you'll die is if you go out too strong. When regen is your friend, time is also your friend. Even if you dont get any kills, you'll still get a big advantage through ap-farming with Baleful Strikes.

Mid game:
Now you are susceptible to ganks (because of low max hp), can always outharrass and should have (in worst case scenario) at least 50 ap. Start harassing with Baleful Strike, hopefully you're opponent will be a bit too aggressive because of his/her/their earlier dominance of the lane and let you land 3 balefuls before getting cautious (if not, harrass anyway :3), allowing you to go in and finish them off with your full combo. Keep farming and do go gank other lanes if there are any good opportunities and if your lane is protected. Go back to base when needed and get your Ionian Boots of Lucidity to increase ap-farming and make you harder to kill. After that, get Rabadon's Deathcap as soon as possible for the nuking capabilities which Veigar was made for.

End game or everything after Deathcap:
Your Philosopher's stones should have paid themselves off about now and is now a personal gold mine of yours which will let you reach all your expensive items very fast. Sell them when you need space for more items. Always have Elixir of Brilliance on you, since teamfighting is all you do now, it will always pay off. I usually play a somewhat aggressive playstyle where I often initiate fights with my stun and full nuke, hopefully putting one enemy out of his/her misery. You deal loads of dmg and should therefore focusing on two things: keeping yourself alive and in range to continue dealing damage and stunning etc. Veigars aoe stun is so effective I often find myself having as many assists as the tank.

After the Deathcap I get a Rod of Ages (RoA) to get you out of your squishyness and after that is where your item build starts to vary depending on your opponent. Now you go for Void Staff if the enemy teams damage dealers have started getting magic resistance (most likely), Rylai's Crystal Scepter if melee champs are chasing you around all the time etc. What you need is more ap, but all ap-items comes with added effects such as +hp/+manareg/+slow/+armor/+mrp etc. Your build should never be set in stone. As end item I go for Deathfire Grasp now. It's the greatest item for raising the amount of hurt you do on one target blasting and can be taken now that you're not as squishy and have a tank and whatnot. With it you have an excess of cdr and can sell your cdr boots for some faster ones, or for some Mercury Treads.

Priority as you may have noticed is:
1. Putting one point in every skill
2. Primordial Burst
3. Baleful Strike
4. Dark Matter
5. Event Horizon
I choose to go for Baleful Strike first as you need to farm as much ability power as possible. Leveling this spell up gives it a lower cooldown and heightens its inital damage which is necessary for both farming and harassing.

Going for Dark Matter midgame is very necessary now as with Philosopher's Stones I can afford to spend mana on killing caster minions with it and also it increases dmg output. I put 2 points in Event Horizon first though so I can land Dark Matter on champions with Mercury Treads.

Veigar has one escape mechanic, Event Horizon. You will need this for offense at all times, only using it for defense if you're getting focused, ganked, is fleeing from a failed teamfight or is saving a teammate. When badly positioned you can hopefully get out of all situations if you do not die instantly with flash and ghost.

When ganked, if you see the opponent racing for you, ghost your way to safety. If you were asleep, too focused on last hitting (I do that a lot :/) or ganked by a flash+taunting rammus or a Warwick, hopefully you can get out of harms way in time with flash. I recommend flashing first and stunning after so its easier to position Event Horizon between you and the opponent.

Also, mobility is King!

Not sure what to put here since I can't find a better way to summarize my build then how my introduction does. I made this guide for you guys and if you have something you wanna ask me about Veigar or ask about the build like "No archangel's gtfo?!" or "All other builds use DFG, u suck?" I'll be happy to answer you if that is necessary. Actually, all comments are welcome regardless (REGARDLESS) of what you say. Try my build, it's awesome. All Q&A will be answered and posted in build. I might add a section on how to counter all champions in game later.

Gl & hf guys/gals!