Please remember that this is just my way of playing him, every game is different and so should your build be.

Pros

Not Item Dependant
Very Big Burst
Can Check the "high grass" from distance
2,5 sec AOE Stun
Good Gank stopper
Good Tank Killer
Can Get infinite AP

Cons

Very squishy
Very mana dependant early game
Need Fast reflexes
Bad building destroyer

Clarity
Clarity are very good to keep you mana pool up, i usually save it for when i'm about level 6, beacuse before that you don't really make enough damage unless you enemies are really dump.
Flash
Flash is a very good summoner spell, i play with flash on almost every champion, you can use flash to jump over walls if you are getting hunted or if you are getting ganked you can flash away plaxe you stun and then you'll probably survive.

Equilibrium - Veigar's mana regen is increased by 0.5% for each 1% of mana missing.

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 (+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.

Cost - 75/75/75/75/75 Mana
Cooldown - 8/7/6/5/4 Seconds
Range - 650

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 (+100% of Ability Power) magic damage.

Cost - 100/110/120/130/140 Mana
Cooldown - 12 Seconds
Range - 900

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.
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.

Cost - 80/95/110/125/140 Mana
Cooldown - 24/22/20/18/16 Seconds
Range - 650

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
Cooldown - 130/110/90 Seconds
Range - 650


Sequence - I max Baleful strike first, as it is a target spell and does alot off damage, as soon as i get to lvl 2 i take my stun. this will help you ALOT with getting kills or flee. when i can't take my baleful strike i take Event Horizon until i got that lvl 3, beacuse before its lvl 3 it doesn't stun enough time so it will be very hard to hit with you Dark matter, flee or setting up a gank, then i max my dark matter, beacuse of the HUGE damage it gives. i take Primordial burst whenever i can, this is just a great skill, it does 500+(100% of YOUR AP + 100% of THEIR AP) this mean that even if you play against a fed ryze or anivia you can just use your Baleful strike and the Primordial burst and they will be dead, or at very low health.

Skill Order
"E" "W" "1" "Q" "R" (Assuming your Deathfire Grasp is placed in your inventory slot 1.)
This will stun your enemy, place your Dark matter making it fall down before they get out of stun, take 30%(or more) of there hp from Deathfire Grasp, hitting them with your Baleful Strike, and ending them up with your Ultimare.

For Masteries i go with a standard 9-0-21 to get you some, AP, CDr, Magic Pen and mana regen.

For runes i go with Magic Pen. wich will help you do more damage and mana regen to keep you mana up then i choose AP per level in both Glyph and Quintess to give you about 50 AP at level 18, the reason i chooser AP per level rather than flat AP is that when you reach about level 6 you will have the same amount of AP as you would with flat AP, so therefore i think it is much better to take AP per level rather than flat AP.

CORE

Mejais Soulstealer - This gives you a huge AP boost and a very good CD reduction, but don't buy it unless you are doing good, if you don't like mejais or ain't doing that good i would suggest you to get Haunting Guise for some AP and HP.

Deathfire Grasp - Deathfire Grasp is such a great item, it gives manaregen, AP and CD reduction and its Active takes out 30%+(3,5% per 100AP) of your enemies current HP, any tanks nightmare to see about half of their HP taken out in 1 single shot.

Sorcerers Shoes - Not much to say about these shoes, they give you movement speed and Magic Penetration wich is great, this way you will do more damage.

Zhonyas Hourglass - This is a new item, it is very good it gives a huge AP boost and 50 armor in 1 item! this mean that beside getting alot of AP you won't be that squishy to physical attacks anymore, and it has the same Active as the old Zhonyas Ring: Places your champion into Stasis for 2 seconds, rendering your champion invulnerable and untargetable but unable to take any actions. 90 second cooldown.

Rabadons Deathcap - This item is your food and water! it gives 155 AP! and increase your AP by 30%! when you get this you will see your AP go from about 500 to 850!

Archangels Staff - now this is your last item, it gives AP and mana regen, and 2% of your max mana is converted into AP, now this will leave you at about 1100 AP!


Other Items
Rylai's Crystal Scepter - Gives HP and AP
Lich Bane - This gives Mana, AP, Magic resist and most important: whenever you use an abillity your next auto attack will deal a bonus 100% of your AP in damage, this will make you 3 hitting towers late game i take this instead of Archangels or Mejais sometimes (only take this instead of Mejais if you ain't doing that good)
Rod of Ages - gives you ALOT of HP (as a caster), good mana too and AP

Early Game
I Usually Head for the mid Lane, but veigar works well in top and bot too. Stay behind your minions, last hitting with your Baleful Strike, don't go to far over on the enemies side as you are VERY squishy. Try to hit the enemies a few time with your Baleful Strike so they always are on half-low HP.

Mid Game
Go and help ganking, but be aware of your lane, you don't wan't to loose a tower beacuse you were greedy. Be VERY aware of ganks, the minute the enemies catch you up in a stun you will be dead! Be VERY aware of invisible enemies, as they can get you killed like 1-2-3.

Late Game
Try to set up some pushes from all side, eventho Veigar ain't the greatest pusher he will help ALOT by stunning/killing the enemies or just scaring them away. The samething about defending.

1. Check you targets items, summoner spells and mana before you are setting up a gank, you don't wan't to end up giving the enemy a dopple kill beacuse you didn't notice (s)he was feeded.
   
2. Use your Dark matter on minions and the Litches, it will grant you a good amount of money
   
3. If placed correctly your stun can stun the whole enemy team.
   
4. If you a getting chased/ ganked flash away and use your stun, eventho the stun doesn't hit they will have to run past it on stop wich will get you in a safe distance from them almost everytime.
   
5. Get the blue buff/dragon everytime you can.
   
6. Never stop last hitting with your Baleful Strike
   
7. Be aware of Anivia, Evelynn or Ezreal. as they can stun you from a good distance (Evelynn goes invisible and stun you, Ezreal can't stun, but he can use his "E" to teleport to you and kill you if you ain't fast with your stun.
   
8. If getting snared by ryze simply place your stun ONTOP of him (so the edge hits him, wich will instantly stun him) before he does any more damage to you.

I Apologize for my bad English and my incorrect use of commas and sentences.
Thank you for taking your time to read my guide, please rate/comment please don't rate -1 without leaving a comment on why you don't like it.
Thanks to DEWO for his Guide on how to create a Build