WARNING! This build is not intended for Veigar beginners. It is a very difficult build to utilize effectively. Veigar is a difficult champ to use in normal situations and this build is only complicating the matter, but it perfects Veigar's nuking capabilities.

First lets take a look at his passive and spells:
Equilibrium - Veigar's mana regen is increased by 0.5% for each 1% of mana missing.

I personally love this passive for Veigar. It makes working up his AP with Baleful Strike a breeze.

Baleful Strike - Unleashes dark energy at target enemy dealing magic damage. If a unit is killed, Veigar gains some ability power permanently.
Active: Unleashes dark energy at target enemy dealing 80/125/170/215/260 (+0.6) magic damage. If the target is killed, Veigar gains 1 ability power.

Passive: Veigar gains 1/2/3/4/5 ability power when he kills a champion from any source of damage.
Cost
75/75/75/75/75 Mana
Range
650

This is your source of power and the extras assigned to this ability are the only reason to play Veigar over other champs. This build makes use of that advantage.

Dark Matter - Veigar calls a great mass of dark matter to fall from the sky to the target location, dealing magic damage when it lands.
After 1.2 seconds, dark matter falls from the sky to the target location, dealing 120/170/220/270/320 (+1) magic damage.
Cost
100/110/120/130/140 Mana
Range
900

Very powerful ability. Hard to hit with, but very powerful. Needs to be used in conjunction with the stun in order for it to hit reliably.

Event Horizon - Veigar twists the edges of space around the target location for 3 seconds, stunning enemies who pass through the perimeter for a short duration.
Veigar twists the edges of space around the target location for 3 seconds, stunning enemies who pass through the perimeter for 1.5/1.75/2/2.25/2.5 seconds.
Cost
80/95/110/125/140 Mana
Range
650

This is a tough ability to get used to. The rang on it is very short if you are measuring the distance of the circle to Veigar, but the opposite edge has a decent distance. Timing and placement take a lot of practice. This ability is too important to ignore.

Primordial Burst - Blasts target enemy champion, dealing magic damage plus the combined AP of Veigar and his target.
Blasts an enemy champion, dealing 250/375/500 (+1) plus his target's AP in magic damage.
Cost
150/200/250 Mana
Range
650

Your Cannon. This will blow up any AP carry like they were a balloon. This a key part to your nuking combo if you want a definite kill.

PLAYING VEIGAR

The traditional combo for nuking with Veigar is to notice an enemies movments, stun and immediately place your Dark Matter shot which is followed up by Baleful Strike and Primordial Burst. This is usually enough to kill a carry without MR so long as you have been farming AP.

Farming AP is what makes Veigar unique. His Baleful Strike pumps up your AP by 1 every time it last hits a target. This ability also pumps up your AP by 1/2/3/4/5 after you kill champs by any means. This allows you to become the most powerful caster in the game and a powerhouse early game if you rock the last hits.

With my core item build the nuking process becomes a little more complicated. First you need to identify your target by category: AP or other. If they fall in the AP category, the traditional nuke combo should still be more than enough. The other category hold quite a few champs that could survive your nuke and that is where Deathfire Grasp comes in. You should try to open up the fight with it which will normally deal magic dmg to them equal to around 40% of their life on average throughout the game. This doesn't seem that complicated until you factor in the fact that they are going to try to get the hell away asap after that hit so you need to respond quickly with your stun and follow up with the nuke combo. If it doesn't scare them and instead they try to kill you and are melee, toss up a dark matter and use Zhonya's activate. It works way to often. After that just ulting them should be enough unless they are very tough tanks or have trick up their sleeves.

After the core items you should more than likely get items that will make you tougher. That is always the case unless of course you get tanks stacking magic resist. In that case you should get a void staff and anything else that suits the situation.

RUNES

RED: Magic Pen always
YELLOW: Mana Regen or Health
BLUE: Magic Pen, MR, or Mana Regen
QUINT: Magic Pen or Flat Health