Hello fellow summoners. I love veigar and he is my main. I have mained him since level 21.
I decided to make a guide, because nobody plays veigar and he's horribly underrated. Besides, most people who do play veigar play him horribly wrong.
This is my first guide, English isn't my main language, and I will make mistakes. You should report my mistakes in the comments. I hate mistakes :p

This guide will focus on my feared insane tank-killing instanuke squishy vaporizing pubstomp build. But I'll try to give some more examples of what you could do. If your positioning isn't that well, or if you're vs a team full of tanky dps (which is veigar's worst nightmare ;( )

Even though I have over 150 matches with veigar. My way of playing him is probably still not perfect. Please do post critizing comments.

Veigar's role is to kill. To instantly exterminate the highest threat of the enemy team. And if you're not too fed (or at least farmed) to at least kill a mage.

In a teamfight, you should focus on stunning as much people as you can. Especially the one certain champion you want to kill. In the bushes west from the golem camp, and south of baron you expect the enemy team to go for baron. You see their huge cho'gath go into the bush in the river. The rest of their team follows him into the bush. Their lux cv's your bush. Oh no, you've been spotted >:(
Your malphite quickly ults in their bush. You place the edge of your stun right into the bush. Now quickly throw a dark matter right into all of their faces and the teamfight is basically won. But just to be sure, ult their annie. Throw a deathfire's grasp on warwick together with a q. This is obviously a very optimal situation.
Your main goal is to shutdown either their ranged ad carry or their mage. Next to that, it's very important to stun as many enemies as you can. This stun wins teamfights, believe me. And your enemy team will ragequit if three of them lost 1/3th of their health because of a well placed dark matter. Veigar is simply extremely useful in teamfights.

Too bad, veigar isn't perfect. He's got his counters as well.
First: Champions that counter him very well.
-Mordekaiser. His shield and his high survivability are just not fun for veigar. After your nuke, he will still have around half health if he is built properly. And since you're so squishy. You will most likely die.
-Nocturne, this is literally veigar's nightmare. He is tanky, he has a spell shield and he kills you in less than 4 seconds. Even if you're not near noc, he will ult you and you are dead. I find it incredibly hard to fight a nocturne.
-LeBlanc in the laning phase. She deals incredible damage and can stop you from farming. You can't nuke her properly because of her sick escaping. You'll nuke her, her passive procs and she simply jumps out of your range. And if you're still too close, she'll cast her ult and either kill you or jump even further away.
-Nasus.. I don't even want to talk about this one, he's the combination between noc and mordekaiser. He will literally one-shot you, and there's simply nothing you can do against it. Unless your team consists out of 4 ad carries without cc he will have mercury threads. And your stun won't stop him. What you can do is just flash over him and as quickly as possible fire all your spells into the enemy team. Your goal is not to survive, but to kill. You will still die for certain, but you did something at least.
-Sivir is probably the hardest to kill 'carry'. With her spell shield she can block your spells. She's definately the least annoying out of these 7 champions but I can't stand it when she simply blocks my 'would have finished her' ult.
-Irelia she is probably as horrible as nocturne. Your stun will stun her for about 0.3 seconds in a 5v5 teamfight, you will instantly die. Unless the enemy is really close, you won't even be able to pull of the trick I told you about at nasus because of her slow.
-This one might sound a bit strange, but veigar will counter himself extremely well.
Since veigar builds lots of ap both veigar's will have say 500 ap. This means that your ult will deal base damage + 1000 damage. This is practically true damage because of my magic penetration fetish. If you're not dead yet another q will suffice.

Items:
-Mercury's threads. For obvious reasons. It's harder to land your w after you stunned somebody. If you're good at aiming w you will be able to hit it. But it's still very annoying.
-Banshee's veil. If it's up. Do not even hesitate fighting that person unless you're sure you can safely q it away. (In a pre-teamfight poking fest for example)
-Quick silver sash and cleanse. Veigar needs his stun to be effective. These are the hard counters for his stun.

Veigar's worst nightmare would be:
Irelia/nocturne with quick silver sash, cleanse, merc's and banshee's veil. There is no chance you will survive this. Literally no way. This rarely happens though (Luckily ._.)

And his last nightmare is:
Extremely late game. Veigar excells in mid and late game. But when everybody has finished their builds everybody is a nuke. Ashe will one-hit you, irelia will 2 hit you. You won't be able to effectively instanuke somebody, because if you get in range ashe will hit you. And you died. Too bad. Veigar is not fun to play lategame. Trust me :(

I usually get flash and teleport.
Flash is a skill I will always get as veigar.

What you could also pick instead of teleport:
Ghost - To escape and catch up with escaping champions.
Teleport - Mostly for extra farming. But if any of your turrets gets assaulted by 3 opponents, and only one person is defending. You could go there. There are next to no people who understand veigar's powers. You can nuke one of them, and the minions would be gone as well with your w. He's incredibly good at defending turrets.

Cleanse - If you somehow get stunned in a teamfight. Or silenced and annie is coming for you, you don't want to stay stunned. Your q + r will kill annie off immediately. Veigar does NOT want to be stunned or silenced. Cleanse is nice for these situations
Clairvoyance - If you don't have a support in your team, you could consider picking clairvoyance. I don't think the second summoner's spells is too important for veigar. And clairvoyance is simply incredibly useful. This is my least picked summoner's spell though.

9-0-21
In offense I go for the ap, cooldown and most important the 15% magic pen. This one really helps a lot.
In utility I pick flash and the upgrade for your second summoner's spell (in my case usually teleport)

If you really don't like being squishy.. You COULD go for 21 in defense, mostly for the ardor mastery. Which gives you 4% more ap. I rarely do this though. But it's not bad.

My runes are fairly simple:
Marks 9x magic pen. This one is pretty obvious.
Seals 9x mana5/lvl. Veigar needs his mana. He always has to have enough mana for his q spell to farm with and his e spell to escape with. You won't have to build extra mana on him. His passive grants him enough mana regen with these runes and death fire's grasp.
Glyphs 9x magic pen. You can pick different glyphs if you want.
Quints 3x ap/lvl. These aren't necessary either. You could replace these and the glyps with flat health.. Or flat ap if you like to be more agressive early game.
Edit: I think I'm quite an idiot, why take ap/lvl quints if you can get the 21 ap with lasthits as well? The extra ap would probably help a bit but it's probably way better to take flat health runes. Or even more magic pen to deal true damage.. Which would be insane with veigar :)

These are my runes.

my feared insane tank-killing instanuke squishy vaporizing pubstomp build: I start off with the mana regen meki pendant and 2 health potions. I get either boots + a 435 ap book. Or sorcerer's boots. I will now rush death fire's grasp. Now I'll start building a rabadon's deathcap. This item grants you insane amounts of ap, which veigar can utilize very, veerrry well. Now I'll either start building an abyssal scepter or a void staff. Void staff when they have lots of magic resistance (>100). Abyssal scepter when they have less than 100 magic res. OR if you really need the extra magic res yourself.
When you're here, you'll deal INSANE amounts of damage, you can take 1/3th of hp of a full RoA stacked malz with your dark matter. Imagine this when you stun 3 of their team, and hit all three of them with your dark matter. They will cry ;(.

Now, you've got 2 slots left.
Items you could get:
- Rod of Ages. For the extra survivability and ap. It's just a nice mage item. You will sustain ashe hits now :)
- Lich bane. This one needs a seperate mastery build though, perfectly explained in this video at 1:39 to 2:15. You don't necessarily need this page, I don't really recommend it either. But you could try this. It works fairly well. Also, don't get the lichbane if you don't have this specific mastery build. It won't work well. Play Video -Zhonya's hourglass. If the enemy team is ad heavy you could consider this item. It gives you a really nice ap boost next to very nice armor and passive.
-Haunting guise. If your farm doesn't go too well late game, this item will help you with a liiittle bit of extra survivability and some magic pen. I really love magic pen as you might have seen :P You could also get it earlier, instead of the magic pen staff.
-Will of the ancients. It's not ideal for veigar himself, but the aura will help your team if it has got multiple mages. It will certainly help
-
-Banshee's veil. I recommend this item for about every champion. It's just very good vs other casters and cc. Only pick this if you're getting killed before you can kill or at least get your full combo off.
-I haven't tried this myself yet, but a combo of lichbane and malady will make you a turret wrecker according to my knowledge. 40% of 500 ap + base damage *1.1 atk speed = turret dead in about 6 seconds with 2 lichbane procs. It should work, I might try this. But I'm not sure if it would work, it's experimental ;p


Deathfire's grasp: This is probably the most important item for any champion ever. I don't get why other veigar players don't get it.
If you're not fed, you'll still be able to nuke down their kassadin or annie. As long as you fire all your spells in this order: Dfg -> stun -> carefully position your dark matter -> q and if he survives that, ult him. And if he survives that (with barely no hp) follow him and cast another q, and even autoattack him if he's incredibly low.
Not only does it do that, but it also gives you mana regen to stay in lane (in combination with your passive, runes and masteries), it gives you some ap and Cooldown Reduction. This is nice as well :)

Also, I will test how well boots of mobility work, I saw this strategy at riot's livestream yesterday evening. I'll test this, and I might add this as another option for boots.

+ Insane, INSANE damage
+ Very nice in teamfights
+ Can easily escape from ganks if done well
+ Can instantly vaporize any squishy or mage. And if fed you might even be able to nuke a tank.
+ Makes opponents ragequit after being nuked.
+ At uncomprehensably late game (after say 5 hours) You can build him full tank, and still have 5k ap.
- Very squishy with my build
- Has quite some counters ;(
+/- Can carry, but because of the easy countering it's hard to carry.
+/- Needs mid for farming

Veigar has some really nice tricks.

Kiting with veigar:
You're farming at bottom lane. Your bottom turret is destroyed. You are alone.
And suddenly (ap) Teemo comes out of the bush. You run away, but step on a mushroom :(
You get down to 1/4th of your health. Haha, teemo has victory. You quickly flash away.
Teemo will follow you, stun him. Get a dark matter off, and hopefully a q (and if you can, just immediately nuke him :P). They usually don't give a damn about that they're at half health now. Now, quickly turn around and cast your ult. Bye teemo. :D

Stun sniping #1
This is probably the biggest mistake most veigars make. An enemy is near. You place your stun right around the enemy. The enemy is not stunned, and is too bad for you. Vladimir. Oh gosh.. You're dead.
Instead of placing your stun around the enemy try to place the edge as close to the enemy as possible. And if you're as experienced with veigar as me, you'll be able to instantly stun him. You can now safely nuke him or run away depending on the situation.

Stun sniping #2
Oh no! (ap tf, you know how this will end >:D) Tf comes to gank you with his ult! Now, instead of running to your tower, simply place the edge of your stun where he'll end up. Throw a dark matter, and q him when he shows up. Throw in an ult and his team will rage at him. Also, try to place your stun in such a way that it will block other enemies as well. This is important for escaping. Since there are most likely two other enemies in your lane as well :P

His nuking!
Don't ever forget that your damage output is insane, if you're getting chased even with only 1 hp, you can most likely at least scare the enemy team off by placing a stun. And if they're dumb enough go in the stun they will soon see the message on their screen 'an ally has been slain' To make sure you won't die you can always just flash away. Veigar is simply amazing at unexpected tides of battles/chasing.

This was my veigar guide. I hope you've learned some new tricks.

I will add more to this guide later. I will also make a veigar kill compilation video. And maybe some graphical examples of some stuff in my guid.

Thanks for reading, and don't forget to comment (rate and subscribe ._.)

:)