Veigar have unlimited AP, he can gain 1 AP everytime he last hit a minion with Q, and 5 AP each time he kills an ennemy champion (even without Q)
His ult is also based on the enemy AP (80% of his ap)

The most important thing to do is to farm Q, a ****in lot !
Don't attack early, you just want to farm your Q and gain more and more AP.

Pros :
Veigar have unlimited ap, so even if a game seems lost, defend as much as you can, farm your Q, and you might have a chance to win lategame.
AOE Stun with your E.
AOE Damages with your W, it deals lots of damages in teamfights.
Can reveal bushes with W.
Funny yordle walk.

Cons :
Really squishy yordle.
All combo is dependent of E; if you fail it, you won't deal so much, and you also can die.
Needs farm to be efficient.
If the ennemy get early Mercury's Treads, your E won't stun enought time for you to land your W until E is lvl 5.
E is quite hard to master in smartcasting, practice and practice it again and you'll be a pro stunner even in smartcasting :)

Ignite and Flash are the best you can take.
Ignite helps you to grab a few kills early, but you won't need it lategame cause you'll OS about everything except full tanks (If you farmed/killed well, it's not even a problem ^^)
You can avoid a gank with Flash, but you can also Flash to stun someone and burst him to death.
The only alternative i can see is Cleanse/ Flash against ennemies with strong cc like Annie, Ahri, Malzahar, Warwick, Galio, etc.

Try to farm your Baleful Strike as much as possible, it's the key to victory.
Don't attack until you deal enough damages.
If the ennemy is pushing minions to your turret, auto attack them when they are near it, and when they're under, you can still farm, kill the one focused by the turret, this is what you have to do when they are full life :
Melee minions : wait 2 turret hits, then Baleful Strike.
Ranged minions : you see the turret focusing him, auto attack once, the turret hit, Baleful Strike.
Tanky minions : autoattack him, and when he's low enough, Baleful Strike

If the ennemy don't push att all to make you come to his turret, last hit with your Baleful Strike when they have the less hp possible. When you are getting too close to his turret, use your Dark Matter on the ranged minions, autoattack them to push a lot and make your minions go under his turret; then go back buying something, and when you come again, they should be near your turret and you can continue to last hit as usual.

Up your E at lvl 2 in case of a jungler's gank (yours or the ennemy)
Up again your E at lvl 5, it allow you to land W, but they can still flash to avoid it.
Always place your cage ON champions, so you have to put one of the side of the cage on them. Otherwise, you'll just block him a few sec, but he can still walk inside and avoid your W or flash over it to chase you.
If the ennemy get Mercury's Treads, you'll have to max your E to be able to land W.

Tear of the Goddess isn't a necessary item if you play with your friends who give you the blue buff; but it's LOL and there are often trolls/failers/*******s. If you don't want to get it, you can grab a few dorans.
Once you get your Tear of the Goddess, you'll have no more mana problems with your passive, you can start to farm your Q non stop. It'll also be really usefull to always have mana in teamfights.

Hextech Revolver isn't always needed, but it helps a lot to get substain against strong harass ennemies like cassiopeia, morgana, etc. With your massive damages, it'll finnaly heal you a lot.

When you farmed a bit, try a small combo E W Q, see how much hp the ennemy lose, then if you think the ult will be enough, do a hard combo E W R Q and make him disapear :)

Once you get your Rabadon's Deathcap, you should be able to burst the ennemy ap carry and here it starts.

After you get your Deathfire Grasp, the combo is a bit different, go for E W DFG R Q, it should be enough to os any ap carry.
Lategame, you can even make disapear the ennemy tank, it's quite good to engage a teamfight :p

When you're full stuff, you can turn the hextech revolver into a Will of the Ancients if you have a heavy AP Team. You can also sell it and buy a Zhonya's Hourglass if you're heavily focused, or if an ad char is fed, or facing a Karthus to counter his Requiem, Vladimir's Hemoplague etc. Buy a Banshee's Veil if you have trouble against Ap Chars or a Blitzcrank, Nautilus, LeBlanc, Brand or someone else who needs combo.

If you have been countered, check what you're facing :
If the ennemy needs skillshots like Galio, Mordekaiser go for Boots and 3 pots. Then, rush Mercury's Treads and Hextech Revolver. If you're facing a Galio, rush Void Staff after the Hextech Revolver. If you're facing a Fizz, rush Banshee's Veil starting with the Negatron Cloak after the Mercury's Treads
If the ennemy don't use skillshots but is AP, like LeBlanc, Ryze, etc, start with Null-Magic Mantle and 2 pots, then upgrade them into Mercury's Treads to reduce Silence, or cage time. Then, rush Banshee's Veil starting with Negatron Cloak and Ruby Crystal.
If the ennemy is AD, like a damn Talon, start with Cloth Armor and 5 pots, then get Mercury's Treads to reduce his silence, and rush Chain Vest to build Zhonya's Hourglass and activate it when he use his ult.

Veigar needs farm to be effective, so you don't want to pick it against every char.

Karthus : You just have to avoid his Lay Waste and farm your Q. Don't get too close not to get slowed by his wall, and you should burst him down pretty easily.

Kennen : Just stay behind minions to avoid his Thundering Shuriken and farm your Q. He often use his lightning rush to farm or to try to harass/stun you; stun him when he does it and combo him. He needs 3 stacks to stun you, play safe if you have 1 stack, and go back under your turret if you have 2 stacks.
If you're having trouble on his stun, get Mercury's Treads.

Ahri : Since Veigar's up, you can now stun her even if she dashs through your cage. If she isn't stunned, delay your burst to make sure you land your W on her and not on her previous position before stun. Try to avoid her 2 skillshots, and especially her Charm. Play passive until your E reachs lvl 3 to insure you to go back after your burst and not get caught by the damn charm. If you're having troubles on her, you can get Mercury's Treads to decrease the duration of the charm.

Morgana : She can't really stop your farm. Just stay behind minions to avoid her Dark Binding, but don't get stuck into minions cause you might take damages from Tormented Shadow. If she harass you with Tormented Shadow, get an early Hextech Revolver or a philosopher stone to get enough substain and continue to farm.
Bait her Black Shield cause you won't be able to stun her if she use it well; run on her like if you would stun her, and go back if she use her shield. She have about 23-15 sec cd on it, so it should give you enough time to stun her. An other way is to Q her when she use her shield; if you farmed well, it should deal enough damages to take it off and now you can stun her.
If you're having trouble on her (she lands good Dark Binding everytime, flash ult you do death, etc), get Mercury's Treads.

Heimerdinger : Stay behind the 3 1st minions to avoid his Hextech Micro-Rockets, avoid his ch-1 concussion grenade and Farm your Q.
You can also land your W or farm your Q on his turrets.

Before a teamfights, when both teams are looking for an engage, you mostly want to place your E to stun the AD or AP carry coming a bit too close. Burst him down and run fast, cause you'll probably be focused by then entire team since you placed your E. Sometimes, just stun and go back behind your tank to let him engage if the ennemies are too close.

Always try to focus the AP carry. Lategame, you'll be able to OS him only with R or R Q. Check for damages before trying to do that.

Launch you're W everytime it's up, they won't dodge it so much during teamfights, or they'll lose a few precious seconds; it's really helpfull in every case.
Use it on bushes when chasing an ennemy, it can save your life many times.