In this guide I will explain what runes, masteries, and items you need to turn Veigar into a walking salvo. I will explain the sequence in which you need to get your items, and the sequence in which you need to put points into each of your abilities.

Pros

Scales great if played right.
Monstrous burst damage
Really helpful in team fights if you manage to stay away from the damage


Cons

If you don't get good farm early game, you can REALLY drop off later
Squishy
squishy
squishy

Greater Mark of Magic Penetration: This is good to deal damage without the enemy having to miss out on any of the fun.

Greater Seal of Magic Resist: Reduce magic damage to help you stay in lane longer.

Greater Glyph of Scaling Ability Power: This will help you scale AP along with your items.

Greater Quintessence of Ability Power: The flat AP boost will help you with last hitting.

Offense
Mental Force : Flat AP is good for last hitting.
Butcher : Also good for last hitting.
Sorcery: Help get your abilities up faster. If Veigar can't use his abilities, he's screwed.
Demolitionist : So you can actually be helpful while trying to take down towers.
Arcane Knowledge : Magic pen. Nuff said.
Blast : Just a little help scaling your AP.
Archmage : Increases your AP by 5%. What could be better?

Utility
Summoner's Insight : Help to get Flash off cooldown faster.
Expanded Mind : Extra mana is always good for a mage. And Veigar needs as much mana as he can get.
Swiftness : Veigar needs movement speed because he's so squishy. It also helps to dodge skill shots in lane.
Meditation : Veigar needs mana regen so he can last hit with Baleful Strike as much as possible.

The reason I don't take Summoner's Wrath is because I don't like what it does for Ignite. I just don't like it. You only get the AP while it's on cooldown, and you only get five.

: Since Veigar relies heavily on burst damage, if your entire arsenal doesn't take them down, this might do the trick.

: Veigar is VERY squishy. If you find yourself in a sticky situation, this could save your life.

: This is also a possibility. It could help you if all your moves are up but you don't have enough mana to use your whole combo. However, if you take it, be prepared for people to whine about it.

Baleful Strike: This is your bread and butter. You need to max this as soon as possible. This makes it easier to last hit minions. When youdo last hit with it, you get 1 AP every time. So you see how important it is that you last hit with it as much as possible. It's passive also makes it so that when you get a kill, you get AP according to its rank. This is your single most important skill.

Dark Matter: You need this so you can harass in lane, farm the jungle on the way to different lanes, and deal massive damage to everyone in team fights. It's also good in mid and late game to eliminate entire minion waves. It's not really important early game, but it's helpful in mid to late game.

Event Horizon: This is good for stopping ganks, assisting in ganks, assisting in team fights, securing kills, and escaping. You need one level of this at level four, one more at level eight, then at 12, 14, and 18.

Primordial Burst: Get this when you can. If you get good farm, get stacks on Mejai's Soulstealer, and have all of your items, this is a hydrogen bomb. It doesn't matter who you hit with it, it's going to hurt. It should almost be a one hit on other AP champions.

Starting Items

: Meki Pendant will give you the mana regen you need to keep you last hitting with Baleful Strike, and will also give you a head start on building Archangel's Staff.

: These will help to keep you in lane longer in case your with someone who likes to harass a lot.

Boots

: These will give you some more magic pen. Wouldn't want anybody walking away from your combo, would you?

Scaling AP Items

: You need to get this early because it takes a long time to scale. It gives you a large amount of mana, and a lot of AP. Pick it up early to get the most out of it.

: This is really good, but you need to either be getting kills on your own, or getting a lot of ganks.

Flat AP Items

: This gives a nice amount of AP, but the AP isn't what you want it for. The active is what you need. The reason you need this is in case you have someone like Volibear causing a lot of trouble in team fights. You can just use this to eliminate a large portion of their health. It's important to use it early, however, because it removes a percentage of their current health.

: This gives you the most flat AP out of any item in the game. It also increases your AP by 30%. And that is GOOD.

The Game Breaker

: It's very important that you don't buy this until after you have all of the other items in this build. Once you have this, you'll be dealing over 1000 damage with all of your abilities, and after every one of them, close to 1000 physical damage. This will make you the turret buster 9000. It will also make you the champion crusher 9000.