This guide is for 2 types of Veigar builds, depending on the situation.
The one on the Left is the practical guide for any type of Veigar in a game of 5v5, essentially, it could also be used for 3v3.
HOWEVER, the guide on the right is the non-practical guide and only to be used if Ex. your team's score has 30 kills and the opposing team has 2 kills and the game has already been going on for an hour. In other words, your team is winning so much that it doesnt really matter what you do anymore, or it does kind of matter but either way the chance of you losing is 1 : 600000



(Sorry I just really like Miss Fortune and Leblanc :P)

Pros

- Veigar has HUGE burst if all his skills successfully hit the target
- Veigar has a pretty large group stun, Event Horizon , which can easily be used to intiate team battles to give your team an advantage, or to put a stop to Nunu's Absolute Zero :P
- Has a nice AOE skill that scales on 100% AP
- Has a way of getting AP without even buying items
- Primordial Blast Primordial Blast scares the **** out of everyone on the other team :P
- Primordial Blast can literally make a tank look squishy when you see 4/5 of their maximum HP drop just from 1 skill
- Event Horizon can be used to help you get away from ganks
- Primordial Blast Primordial Blast also uses a percentage of the enemy's AP :D


Cons

- Veigar is very squishy :(
- Veigar's Event Horizon is easily avoidable/countered
- Dark Matter is also easily avoidable
- Veigar's movement speed sucks :(
- Veigar is very dependant on his Event Horizon due to the fact that he needs the skill to run away, use combos (Combos can also be used without Event Horizon, but its much easier with the stun so your Dark Matter can succesfully hit the target), catch runners, gank, and stop channeling skills such as Nunu's Absolute Zero , or Janna's Monsoon .

The Impractical guide is used when you are winning and its virtually impossible for you to lose. You should sell your items and buy the items for the guide, not for reasons of winning, but because its much more fun to watch the opposing team get pissed off at how much HP 1 of your Dark Matter can take off :P
My impractical guide should result you in having 1000+ AP if you have been using enough skills to help your Archangel's Staff increase your Maximum MP.
And since Dark Matter scales on %100 of your AP, it would do 1320+ damage (Magic Resist and Magic Pen not included. 320 at skill lvl 5 + 100% of 1000). Now, your Baleful Strick would do 860+ damage (Same as Dark Matter, MR and M.P. not included. 260 at skill level 5 + 60% of 1000). As for your ultimate, it should do roughly around 1700+ damage (Same as Dark Matter and Baleful Strike. 500 and skill level 5 + 120% of 1000).

Now, 1700 + 1320 + 860 = 3880 (M.R. and M.P. not included)
3880 is pretty good damage don'tcha think? :3

Though this is really heavy damage output from a champion with 1 combo, keep in mind, it is still an impractical guide because using these items won't give you any other items that increases Ex. your Magic Pen like Void Staff , or help you slow the enemy like Rylai's Crystal Scepter . However it is very powerful.

The practical Veigar build allows for some Magic Pen, helping Veigar deal with those tanky champions.


The Impractical guide is impractical... o.o this is a pure AP build, very good for Veigar, but still impractical :x The item order is like that so you can maximize your AP amount by the end of the game.

Practical- Getting Archangel's as fast as you can allows you to maximize the max MP bonus passive from the item at the end of the game.
Ex. Veigar 1 vs Veigar 2.
V1 got Archangel's at 10 minutes as his first item. V2 got it at 25 minutes as his second item because he used his money to get a Rabadon's Deathcap first.
assuming each veigar uses 1 skill every minute and the game ends at 50 minutes and every skill used gives the player 1 bonus AP...
V1 would end up with 40 bonus AP while V2 only ends up with 25 bonus AP.

Also, Mejai's Soulstealer is purely situational, IF you are doing very well, go ahead and get it. But if your doing horrible with a score of something like 0-15-3, then I wouldn't suggest you get it :P
Rod of Ages is also replacable if you feel you have a better item to get. I would get it to help me survive better because Veigar is a pretty squishy champion, but getting it is up to you.
Void Staff last because people tend to finish up their items later in the game, so you won't have to rush Void Staff yet until the very end, therefore you should get Rabadon's Deathcap first. Just because its more effective if you get Rabadon's Deathcap first, where tanks still may not have their M.R. item yet.

Baleful Strike:
- Use this to get last hits for bonus AP
- Pretty spammable so use it to harass in the later levels
- NOT recommended to be spammed before level 6 as it can burn off your mana very quickly
- NOT a huge burst damage skill, don't think this skill is going to scare anyone unless it does 1k damage


Dark Matter:
- Heavy AOE damage, use this to clear minion waves for gold
- Use this on people who are in your way as it will probably scare them away
- Used with Event Horizon increases the chance of this skill actually hitting the opponent, rather than missing and hitting the ground
- Due to the delay of the meteor (or whatever it is that drops down from the sky), try to aim it where you think the opponent will run towards


Event Horizon:
- Big group stun
- Can be used to initiate team fights or just any normal fight
- Can be used to run away, gank, catch other champions who are trying to run away, and stop channeling skills.
- Using this skill for dumb reasons such as preventing minions from reaching your tower, is NOT recommended due to it's long cooldown, and while its on cooldown, you may easily get ganked and have no escape
- When using this skill, place the edges of the blue circle (that you see when you press E, like a skill shot kind of thing... but it's a circle... ._.) onto champions channeling spells, so the stunning part immediately hits them, therefore stopping their channel.


Primordial Burst:
- In teamfights, should be used to take down those heavy nukers, incase they nuke you or your teammates down, such as Annie or Leblanc
- Since it uses a percentage of the enemy's AP, use it on AP heavy champions like...
(Gonna name my favourite casters here :P)
Cassiopeia (Yeah I like using her :( so what? D:)
Annie
Leblanc (She's cool k? :3)
Karthus
Katarina
Lux
Miss Fortune (I KNOW SHES NOT A CASTER :( I JUST LIKE HER K? T^T)
Morgana (Hehehe funny dark ball that does ****loads of damage and comes from behind the wall :D)
Shaco (AP Shaco is fun :) if you go pure AP, your clone does like 2k damage when it dies :D)

Practical - Start off the game in any lane you want. Mid might not be best for Veigar due to his early game weaknesses (Lack of MP / heavy burning of MP, and horrible damage output until Primordial Blast Primordial Blast is acquired), therefore he could easily be zoned away from EXP and last hits, which Veigar is most effective with due to the effect of Baleful Strike . (For every last hit you get with Baleful Strike, you get 1 permanent AP, which might not seem like a lot, but if you keep last hitting, you could get well over 100+ bonus AP acquired from Baleful Strike).
Other than that, keep last hitting with your Baleful Strike. DO NOT ENGAGE IN FIGHTING unless you are level 6! or if your laning partner is getting ganked or something like that, but if you have nothing to do, don't try to harass the enemy. Why? Because Veigar is extremely mana-hungry early game, so if you keep using skills to harass your enemy, you will be constantly recalling for MP, therefore missing out on last hits and EXP. Also, your skills do horrible damage Early-game anyways.

Mid/Mid-Late game... this is where Veigar gets very scary for the other team. 1 combo at lvl 11 should be enough to kill a squishy and severely damage a tanky champion.

Late Game, 1 combo has the potential to almost, if not kill a tank, and almost enough damage to take out 2 squishies. Don't be afraid now as long as you aren't 1v3+


Impractical- Early game: Same as practical ^

Mid/Mid-Late game... Scary Veigar :( Kill as much as you can... no seriously, LOL. go for the squishies, go on a rampage, it doesn't matter if you laugh like a maniac in real life while you pwn the squishies, the more you kill, the better.

Late game, well if your team is competent and hasnt been feeding the opposing team's Tryndamere for 20 kills (or something stupid like that), then you should have no problem killing most of the opposing team (Unless its like... Rammus, Amumu, Shen, Nunu, and Blitzcrank with full Magic Resist items.. o.O)
But yeah, seriously, have fun killing tanks as if they were squishy lvl 1s. :P

Yeah, so... yeah... ._. I really don't know what to say... o.o

Umm... k.. so......................

I hope my guide helped you o.o and... hope it ... yeah... o.o
-Bad at Summaries-
Oh fk... i just realised ive been calling Primordial Burst ... PRIMORDIAL BLAST THE WHOLE TIMEEEEE .... sorry :(
Yeah Miss Fortune is a caster now :)
and yeah Cassiopeia is pro, I like her :O spam is fun :)
Why are there no Leblancs anywhere? T^T SHES SOOOO FUNNNNN D: but no one uses her :(
You know how fun it is to have a triple flash with Leblanc? (Flash, then use Distortion, then use Mimic) or just keep juking the enemy with Distortion and Mimic? :(