Hello, and welcome to my Veigar build. This is the second build I made, the first one being an Udyr build. You can check it out here: Udyr - Offtank Pwnage
Hope you enjoy and learn something from this guide.

Pros
+ Can nuke a single target from level 4~5
+ Only champion in the game who can one-hit enemies. ( Primordial Burst)
+ Build-in and improved Mejai's Soulstealer by last-hitting minions with Baleful Strike and killing champions.
+ Can farm minions easily throughout the whole game.
+ AoE Stun ( Event Horizon) Cons
- He gets focussed quickly
- He is squishy pretty much the whole game

You can go two ways with the runes. The offensive and the defensive way. I'll go over both of them.
Greater Mark of Magic Penetration - These are very basic for any AP caster. e.g. Annie or LeBlanc. They let you do more damage so why not take them?

Greater Seal of Ability Power - This gives you an early game advantage with just a little more AP. I prefer the early game AP runes over the late game with Veigar because late game you can have 1000+ AP when heavily farmed and fed with kills. Then the 10+ AP from a late game runepage doesn't matter anyways.

Greater Glyph of Ability Power - These are chosen for the same reason as the Seals.

Greater Quintessence of Ability Power - These are chosen for the same reason as the Glyph's and Seals.

Greater Quintessence Of Desolation - Just a little more Magic Pen can help you do that tick more damage to kill them. These are thus chosen for the same reason as the Mark's.

Greater Mark of Magic Penetration - These are just necessary, you need some Magic Pen. from the start off. And you would waste your offensive runes by buying something offensive.

Greater Seal of Armor - These give quite a bit of Magic Resist, and it definately helps surviving the enemy AP carry's nukes. Especially early game.

Greater Glyph of Magic Resist - These give a bit of armor against those AD carry who could ruin your game.

Greater Quintessence of Ability Power - Just a little boost in AP for the early game as you are already well prepared for whatever enemy you'll find on your lane.

Greater Quintessence of Magic Penetration - These boost your Magic Pen and let you do more damage throughout the whole game.

Greater Quintessence of Health - These are for the extremely defensive runepage. Veigar is quite squishy throughout the game so these can help you with that. (These almost only make a difference early game.)

The masteries can go two ways as well. Again, the offensive and defensive way. In the offensive masteries, go 21-9-0. Take All the AP/Magic Pen masteries, take full Magic Resist in defense, 1 in the armor, and go for the Veteran's Scars for some early game survivability.
When going defensive, go 9-21-0. Take Magic Penetration in Offense, and then go down the defensive mastery tree.

Core Build Early Game: Kage's Lucky Pick
The Doran's Ring gives you almost everything you need early game.
+ Ability Power
+ 100 Health
+ Mana Regen Veigar is quite mana hungry, this is kind of solved by his passive Equilibrium but not completely so this helps too.

Then you immedeately rush your Kage's Lucky Pick
What are the pro's and cons of doing this?
+ You'll probably earn this gold back because of the passive (Gain 5 gold per 10 seconds.)
+ You have some more early game Ability Power.

- You don't have your boots yet and are thus probably slower then your opponent(s) in lane.

Sorcerer's Shoes
This is an obvious choice for Veigar but I'll discuss it anyways.
+ Gives Magic Pen
+ The movement speed you lacked earlier

Note: If your early game is not so good, and the enemies have a lot of CC, you could consider buying Mercury's Treads it is good for the survivability but your own nukes will start lacking in damage.

Mid Game Core: , &

Why rush my Rabadon's Deathcap, when I have barely any other Ability Power Items?
+ You get a stunning 140 bonus Ability Power
+ 30% increase in Ability Power Yes, we actually do. Right now if you are doing well, you could have ~70 last hits with your Q, which means that you have 70 + 20 ( Kage's Lucky Pick) + 15 ( Doran's Ring) + 140 ( Rabadon's Deathcap) = 245 AP! This means we get 70+ Ability Power from the passive, which is about as much as a Needlessly Large Rod. Worth getting now is it?

Take Rylai's Crystal Scepter on a burst champion? Is that a good idea?
To be honest, it is not the perfect item, but you can better buy this as it has proven me useful. Imagine a tanky Sion stuns you and turns on his ultimate. ( Cannibalism) What would be your reaction? You would probably stun him with Event Horizon and run away. But then again you could also throw your Dark Matter onto him, and as soon as he starts chasing, he is way to slow to catch you.

The Deathfire Grasp is a perfect item for Veigar as it synergizes perfectly with my Q ( Baleful Strike) and my R ( Primordial Burst)

Late Game Build: &

Why should you buy Zhonya's Hourglass?
+ It gives you armor against those pesky fed AD carry's. e.g. Ashe or Caitlyn.
+ When you suddenly get Rocket Grabbed into the enemy team by Blitzcrank. You press 5 and let your team come to the rescue.
+ It gives 100 AP which is really good for an item with such an amazing active.

Passive

This is quite a good passive for early game. It regenerates mana based off your missing mana. It works the same as Chalice of Harmony). This scales too poorly into late game, and will not matter so much then anyways. It does help early game so I guess it's an OK passive.

Q

This skill is AMAZING! It is your harrassing tool, your Mejai's Soulstealer and your finisher. You get 1 Ability Power per last hit, and 1/2/3/4/5 AP per champion kill, depending on the rank of the skill. I can't stress enough how important last hitting is with this skill.

W

This skill will be a farming tool mid-late game, and you can easily combo this with Event Horizon. It takes 1.2 seconds to fall down, so you launch your Deathfire Grasp + Baleful Strike and then your Dark Matter will most probably finish the job. When available, always throw your Primordial Burst after your Q, for maximum damage.

E

Excellent, Excellent spell. An AoE stun which lasts for 1.5/1.75/2/2.25/2.5 seconds depending on rank. It is the starter of your combo, and will keep them in place long enough to give you a free kill.

R

This spell is plainly amazing. It deals 250/375/500 (+120% of your current AP) (+80% of the targets AP) I absolutely love this spell. It is the only spell in the game which can one-hit champions. It feels so good when you see a full HP Lux walking around under her turret, you fire your Primordial Burst and she dies in one hit. The enemies don't even understand what just happened.

Skill Combo's

Harrassing very early game

For free harrassment use E>Q and run. They are stunned so can't do anything back.

Harrassing early-mid game

You can step it up a notch by doing the E>W>Q combo, which should deal a nice amount of damage, and a second burst will most likely kill them.

Harrassing late game

There is barely any way to harrass late game. Just focus one person and use E>W>1( Deathfire Grasp>R. If they live a Q should finish it off. If you feel like you can easily kill them, a combo of E>R>Q can kill squishies/AP Carries.

+ Ignite - This can secure kills early game, and be fatal against lifestealers like Sion or Swain.
+ Flash This spell is, even after the nerf still really useful. Flash over walls to chase, get away or join a teamfight. This is a free escape mechanism too, which is something Veigar lacks.
+ Cleanse - A nice spell, when you get focussed with CC, pop this and Flash out. But it might be better to buy a Quicksilver Sash
+ Exhaust - Not so useful on Veigar as he is a nuker, and 1v1's hardly exist. It is ok to shut down an AD carry in a teamfight or chase with the movement reduction.
+ Ghost - Oh, Ghost, this could be useful. Not really, maybe when chasing or being chased, but CC negates this. It's ok when you haven't unlocked Flash yet.
+ Heal - Could help when you feel uncomfortable and are still learning Veigar. But I don't recommend it.
+ Clarity - Some people swear by this because Veigar is mana hungry early game. I don't recommend it as it is useless late game.
+ Teleport - Can be useful to teleport back to lane instantly and to get to large minion waves mid-late game. I am not a huge fan of this spell. Your choice.
+ Surge - Meh. A decent pick, it doesn't help at all late game, I don't like it that much, should be ok though.
- Revive You shouldn't be dying, and this is, in my opinion a useless spell.
- Smite - Jungling is a straight no for Veigar.
- Promote - It's not so useful anyways.
- Clairvoyance - This is a support spell, don't take it.
- Garrison - This is a tank spell, and Veigar is not such a great choice in Dominion anyways.

Laning Partners!?

Veigar is primarily played mid, and when playing ranked you should probably do that, but in normal games an AD carry can go mid, and Veigar synergises well with certain champions to play with.

Tiers

Tier 1 (Perfect)

+ Soraka - I love to lane with a Soraka as she provides HP, mana and a possible silence. She can also help you farm minions if needed.
+ Alistar - His CC is quite nice and he has a good sustained heal which helps Veigar quite well.

Tier 2 (Good)

+ Janna, she provides a knock-up and a shield but lacks HP regen and mana regen, this can be solved by buying Mana Manipulator though.
+ Taric He has a stun and a heal with which he can both help killing enemies and sustain Veigar.

Tier 3 (Horrible)

- Other Carries. Don't do this, as you will "KS" each other and need to share kills. This generally doesn't work out well.

Veigar is not very farm dependant, but when he farms, he gets ridiculous amounts of both gold and Ability Power. ALWAYS try to last hit with your Q Baleful Strike, as you get 1 AP. You might think that this doesn't help so much, but I had 260 AP just from last hitting in 1 game. To put this in perspective, that is about 3 Needlessly Large Rods + an Amplifying Tome, this equals 3*1600 + 435 = 5235 worth of items just by last hitting, that is ridiculous!

Mid-late game, you can clear groups of minions with your Dark Matter, even though you won't get Ability Power, you will get the gold quicker.

Comming Very Soon!

Overall, I feel like Veigar is a very fun champion to play with and he can carry games quite well. He is well worth the 1350 IP he costs. I hope you enjoyed my build, and please leave constructive criticism in the comments. Vote please!

Change Log

12-2-2012 - Guide Published!
12-2-2012 - (Update 1.01) Took Zell's advise, and changed some errors ( Greater Quintessence of Desolation --> Greater Quintessence of Magic Penetration) and added Deathfire Grasp in the item section because I forgot to add it there. Thanks to Zell for mentioning it.