Pros/Cons

Pro

-heavy nuke
-nice farming
-incredible AP
-nice Mana/Life late game

Cons

-bad life early game
-a bit squishy

Equilibrium
Veigar's mana regen is increased by 0.5% for each 1% of mana missing.
This will really help you early game.
Baleful Strike
Active: Unleashes dark energy at target enemy dealing 80 / 125 / 170 / 215 / 260 (+60% of ability power) 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.

Cooldown 8 / 7 / 6 / 5 / 4 seconds | Cost 75 mana | Range 650

IMO Veigar's best ability because of the additional AP.
Dark Matter
After 1.2 seconds, dark matter falls from the sky to the target location, dealing 120 / 170 / 220 / 270 / 320 (+100% of ability power) magic damage.

Cooldown 12 seconds | Cost 100 / 110 / 120 / 130 / 140 mana | Range 900

Incredible damage late game.
Event Horizon
Active: 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.

Cooldown 24 / 22 / 20 / 18 / 16 seconds | Cost 80 / 95 / 110 / 125 / 140 mana | Range 650

Nice for escaping, start fights and killing enemies.
Primordial Burst
Blasts an enemy champion, dealing 250 / 375 / 500 (+100% of ability power) plus his target's AP in magic damage.

Cooldown 130 / 110 / 90 seconds | Cost 150 / 200 / 250 mana | Range 650

Nothing left to say, damage, caster killing just nukey

I chooseGreater Mark of Insight for Magic Pen,Greater Seal of Clarity for Mana Reg, 2xGreater Glyph of Focus 7xGreater Glyph of Celerity (you can also take you 9x Celerity/Focus) orGreater Glyph of Force (a bit useless because of your Q but possible) and someGreater Quintessence of Fortitude for flat HP.

I take 9/0/21 caster build withGood Hands not Perseverence and Haste for tier one. Then go Awareness with two points and Expanded Mind with four points at Tier 2. Then take ManaReg, Flash, Movement speed, CDR and Summoner cooldown.

For Offense tree take AP, CDR and Magic Pen

You start with aMeki Pendant for early ManaReg and spamming you Q. You should also buy 2Health Potions. After you got the money buy aTear of Godess . Now you really should buy a pair ofSpeed Boots . Now you gonna get the most important item for veigar aDeathfire Grasp You can start with aFiendishing Codex and getSorcerers Shoes beforebut it's like you want to. Thewill so much improve youre nuking ability and you should use it as often as possible. Now you can decide what to get next. You can decide but here are some examples.
Rylais Crystal Scepter Gives good AP and HP.Mejais Soulstealer Very good AP and CDR but don't buy it if you are bad.Rabadons Deathcap Incredible AP NUKE!NUKE!Zhonyas Hourglass Veigar is so squishy and this item will help a lot vs. CrCVoid Staff Some nice Magic PenetrationAbyssal Scepter Same reason as Zhonyas.Archangels Staff
Generally good caster item espacially with a hight amount of mana and a great need of it like Veigar.
These items are just my opinions, you can take any item you want but i think if you chose a combination of these you will get very good results.

I think Veigar needs some defensive summoners because he is so squishy.
I'm not going to list all of 'em just the one's you should take and not take.Flash Very good for escaping, just flash out and you're away.Ghost Also for escaping out of some situations in which you don't want to rest.CleanseDefnesive spell get away from stuns and some things like that.Heal Also possible to get but not the best chose, cuz late game you'll escape every time.,,,,, never get these spells you won't need 'em they are

USELESS

You have to farm really much with yourQ. If you aren't able to last hit minions Veigar's not the right Char for you. After you got enough AP always shoot a Dark Matter on the ghost beneath the mid lane.

Start with ato stun enemy, after that immediately nuke 'em up with aand now you can wether kill him withor shoot someand use thefor nuking him away.