Hey, I'm Sullii. This is my first guide so be critical on me!
In my opinion, Veigar is a great AP mid, he has a powerful burst combo which is usually considered OP
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Veigar

PROS

Great Ap Mid
Capable of doing a great amount of damage to a single target
Very powerful against AP carries
Ulti is able to devastate and nuke an opponent
He has a leprechaun skin...

CONS

Squishy as hell.
Early Game Burst is fairly weak.
Event Horizon may be hard to land for newer players, therefore may have to rely on either Flash or Ghost

http://www.mobafire.com/league-of-legends/rune-page-planner#&rune=275:9-268:9-272:9-338:3
I prefer this rune build as it gives Veigar the upperhand in terms of damage in lane. The Greater Seals of Replenishment also allow him to spam his abilities more.

Alternate Rune Build

http://www.mobafire.com/league-of-legends/rune-page-planner#&rune=338:1-351:9-319:5-262:9-266:2-263:4

This Rune build is for the players who prefer to play a safe Veigar. Replacing 2 of the Potency Quintessences with the Fortitude ones allow a higher rate of survival from a Enemy Jungler ganks/ opposing Mid Champion. Replacing the Mana Regeneration Seals with Armour Seals also helps sustain as it protects you from taking as much damage as you would in lane. Finally, the 4 extra Magic Resist Glyphs serve to weaken the damage of an early Mid Burst.

The standard AP masteries 21/0/9 can be used but i prefer to use 21/6/3 as it gives me more of a balance.

Alternates

: 21/9/0 could be used for more damage resist and lane sustain.

Recommended

Flash Ghost Ignite Teleport
Flash: Flash is important for Veigar as if for some reason you miss your Event Horizon you may need it. Taking flash will save you many times :).
Ghost: Read Above ^^^^^^^^
Ignite: Ignite can be taken as it can allow you to finish someone off after a full burst.
Teleport: Teleport is a useful spell that will allow you to get into a teamfight instantly and also allow you to get back to lane instantly, preventing the Opposing AP mid to outfarm you.

NOT RECOMMENDED

Clairvoyance Surge Revive
Clairvoyance : Pretty sure Veigar won't need to support.
Surge : It may be effective as it supplies the user with AP but it's not viable for a burst mage
Revive: Maybe Karthus but not Veigar.

Baleful Strike - Q - Baleful strike is one of Veigar's most important abilities. It should be maxed out as soon as possible to maximise AP gain and damage.

Dark Matter - W - Dark Matter is a Hard to Land, High Damaging Skill. Should always be used together with Event Horizon.

Event Horizon - E - Event Horizon is Veigar's only form of CC, it is also used to effectively set up Veigar's burst. (Is Also an effective defence Mechanism. [If landed Properly])

Primordial Burst - R - Primordial Burst is Veigar's "SUPEROP" Nuke. It deals more damage to champions with higher AP.

THE BURST

Event Horizon Dark Matter Baleful Strike Primordial Burst
This Burst is used to maximize damage as the Event horizon Stun is a great setup for the rest of the Burst (2.5 second Stun at max level) . Dark Matter is followed after the stun as it assures a sure hit on the target (320 Base Damage at max level). Next Baleful Strike (260 damage at max level) and Primordial Burst (500 damage at max level) are used right after each other to deal massive amounts of damage.
At minimun if all hits are landed that would deal about 1080 damage. (Not accounting for items and Baleful Strike Ap bonuses.

Always Last Hit when possible, and try to maximize use of Baleful Strike's passive to gain more Ability Power.

Laning Against

Ahri

Avoid her Charm as it is a setup for her burst.
She can jump through your stun with Spirit Rush.

Akali

Has a high chance shut you down at Lvl 6 when she has her full combo.

Anivia

Dont forget about her passive Rebirth.
Try to avoid her Flash Frost as it is a setup for her burst.

Annie

Her Burst is often underestimated but it does alot of damage.
Summon: Tibbers is scary :(

Brand

Try to avoid his Pillar of Flame as it gives him the ability to stun you with his Sear.
Brand is one of the more tankier burst mages.
Cassiopeia

If up against Cassiopeia take boots for extra mobility to avoid her poisons.
Stay away from her Miasma clouds.
Her Ultimate Petrifying Gaze is deadly and can allow enemy jungler an easy kill.
Avoid getting poisoned as her Twin Fang skill loses 0.5 seconds cooldown.

Diana
Try not to get too close as a quick Moonfall and Pale Cascade may severly injure you.
Be wary of her Lunar Rush as it may close distances allowing an easy kill.
<INCOMPLETE>

I don't recommend it but its been done...
Credits to Saintvicious Play Video

26/08/2012- fixed up Skill Sequence. Re-Done Item Build. Reviewed "The Burst".
27/08/2012 - Added Brand, Cassiopeia, Diana to the "Possible Lane Situations" section.

Thanks For Reading My Veigar guide. I will develop it further and add more sections once i get enough free time! :)