I developed this build over the first week I played with Veigar and is my primary build for him still, although I change it in small amounts from time to time. The goal of this build is to make a good laning Veigar (it has not yet been tested for midding or jungling or what have you) that can blast apart most other champions with ease. He also should never run out of mana.

Side note: the runes section has been largely thrown together for now.

The masteries for this build are designed to increase a lot of his stats by a modest amount, with extra focus on regens.

For low-level summoners: the offense column is the most important here, try to fill it out first. Sorcery and Archaic Knowledge are the most important masteries in the entire build, so really focus on getting them. Veigar has pretty high natural cooldowns, mostly due to the massive damage his abilities can do, so getting those down can mean the difference between no kills and getting fed, which is why Sorcery is so important. Having the 15% magic penetration right off the bat that you get from Archaic Knowledge really helps take out low-AP tank champions that would otherwise shrug off a lot of Veigar's damage (esp. his ult).

The next most important is the defense. The items in this build pretty much ignore armor and magic resist, relying on the defense masteries to beef that up a little bit (though he'll still be a bit of a glass cannon, you should have no problems killing people before they hurt you at all). The masteries increase both defenses by 6 as well as increasing health regen by 26.4 at 18 (which is helpful for when you barely escape something, this also comes more in handy if instead of getting Shurelya's Revelrie you get Warmog's Armor or some other large max health booster).

Utility is what you should get last (once again, that's for low-level summoners). Since the utility column mostly focuses on regen and mana, and that's covered with items, it only helps so much.

That all being said, Haste and Cripple should be the first 2 masteries you get, as they can really help you get early kills or keep enemies from retreating after a failed attack.

The items were mostly chosen to increase mana, mana regen and ability power, that being said, the spell vamp is EXTREMELY important, especially if you replace Shurelya's Revelrie with Soul Shroud or Warmog's.

The mana and mana regen from Archangel's Staff is really important early on, as that coupled with Equilibrium can keep Veigar from ever running out of mana (within reason).

As for replacements...

Shurelya's Revelrie is the most obvious candidate for replacement. Shurelya's gives Veigar health and both regens, it also gives all nearby champions cooldown reduction, something indispensably important for Veigar. Its active can really help in retreats, ambushes and pursuits. If you're one who doesn't like actives because there's no way to quickly activate them, but still want a similar effect to Shurelya's, I'd recommend a Soul Shroud as replacement. Soul Shroud gives 90 more health, is a little cheaper and only has passives. Drawbacks: no health regen, less mana regen and slightly less ability cooldown reduction. Or, if you just want to do more damage, and still want some extra health to offset Veigar's low defenses (and naturally low health) might I suggest either: Rylai's Crystal Scepter, Rod of Ages or Haunting Guise. Rylai's: pros: a lot of health increases, pretty good AP increase, and the passive of slowing enemies' movement speeds when you hit them with spell damage, cons: price, movement speed reduction is reduced for AoE spells (like Veigar's all-important Dark Matter). Rod of Ages: pros: good health and mana increases, heals on level, long-building increases towards health mana and AP, cons: while after 10 minutes of having the Rod, it has really high bonuses, it's price means you'll get it late-game, meaning you may not reach the 10 minute mark, doesn't heal by much, no overly unique passives, price. Haunting Guise: pros: price, increase to magic penetration, cons: low bonuses, no unique passive.

I chose Exhaust and Ghost for a couple big reasons.

1. The build doesn't increase Veigar's move speed until later than most.
2. Exhaust with Cripple upgrade can bring kills, especially in the beginning.
3. Exhaust and Ghost used together can give enough of a speed advantage to catch up to and snare much faster champions.

Farming, especially gold farming from minions, for this build is really similar to any other Veigar really. Following the item purchase order, Rank 3 of Dark Matter can kill the spellcaster minions about 75% of the time (it depends on how many it hits), Rank 4 will kill them 99% of the time and kills the melee minions about 85% of the time, Rank 5 will flat out kill any of the two basic minions, regardless of how many of them it hits. This makes Veigar really fantastic for 4 things.
1. He can get a nice gold advantage by blasting the hell out of minions at every turn.
2. He can be great for late-game base defense, as he can take out several minions with one attack, meanwhile focusing on other champions.
3. For the same reasons as 2, Veigar is a really good late-game base attacker. Blasting away at the minions lets your minions build up, and, once again, he doesn't need to focus on the minions, just use Dark Matter from time to time, and since it has a delay, he can cast it then immediately turn and attack champions, or even run for cover.
4. Late-game he becomes a pretty good push and carry, he can blast away all the enemy minions in a lane by way of an Event Horizon - Dark Matter combo. With spell vamp and high hp regen, he'll stay alive pretty well too.

Veigar of any build can be extremely useful in team battle situations because of Event Horizon and Dark Matter. Shurelya's Reverie can also give the entire team a 40% move speed increase for 3 seconds, which is really useful in retreats. Will of the Ancients is not only cheaper and more useful to Veigar than the Hextech Gunblade, but also gives some AP and Spell Vamp to all nearby champions, which can really come in handy in team battles. Veigar can also be used to help kill junglers, as Event Horizon is the size of the entire walkway in the jungle.

This particular build is best suited being alongside a tank, especially since he'll have low movement speed for quite some time. Laning alongside Shen is generally a good idea with this build. I sincerely suggest that you NOT lane with a Janna. While it may seem like a good idea at first, and a way to really insure that you'll get all the kills in that lane, it doesn't work out. UNLESS you and that Janna are very well coordinated, in which case it could work out spectacularly. This is mostly because of Janna's affinity for moving enemies, while Dark Matter, Veigar's serious damage dealer, needs enemies to be in a specific location. Janna's pushing can also push enemies through Event Horizon, causing them to not get stunned by it, and becoming immune to it (if someone physically passes through an Event Horizon without getting stunned, for any reason, they become immune to being stunned by that particular Event Horizon, for instance if someone used Banshee's Veil, and walked through EH, then their Veil would block the stun, and then they can walk through that EH again and won't be stunned this time either, this is largely a glitch caused from EH being unable to double stun the same person).