Hello everyone and welcome to my guide!

In this guide, I will be discussing how to play one of my favorite champions, Veigar, The Tiny Master of Evil.
Veigar is one of the fun champions to play in LoL, and in my opinion, the best AP Carry, as he does not cap on damage. He's able to hit for insane amounts with his ult, 2k+ depending on the champion.
This little guy was the first champion I ever played, along with the first champ I fell in love with.
Well what are we waiting for? Lets jump into the guide!

Runes are very important on Veigar. He still hits hard without them, but with runes, Veigar hits like a truck.
I like to use:

Marks: Magic Penetration(Greater Marks of Insight) x9
Seals: MP5 @ 18(Greater Seal of Clarity) x9
Glyphs: AP @ 18(Greater Glyph of Force) x9
Quints: Flat AP(Greater Quintessence of Potency) x3

I use this set up because

The marks give you magic penetration, so you cut through some of the set magic resist all champs have, allowing you to do more damage early/mid game.
The seals give you some extra mana, so you don't oom as fast.
The glyphs give you some extra end game damage.
The quints give you some extra early game damage, to help last hit minions with your q, allowing you to get even more ap.

Other viable Runes include

Greater Mark of Force(AP @ 18) - these give you more end game ability power, meaning your abilities will hit harder.
Greater Mark of Potency(Flat AP) - these give you a better early game, which makes your farming with q easier.
Greater Seal of Force(AP @ 18) - these give you more end game ability power, meaning your abilities will hit harder.
Greater Seal of Potency(Flat AP) - these give you a better early game, which makes your farming with q easier.
Greater Seal of Replenishment(Flat MP5) - these give you slightly more MP5 before level 7, before the MP5 @ 18 seals even out to these, which makes you more capable to spam spells early.
Greater Glyph of Potency(Flat AP) - these give you a better early game, which makes your farming with q easier.
Greater Glyph of Insight(Magic Pen) - these give you some extra magic pen, which lets you go through some more of their magic resist, allowing you to hit harder.
Greater Glyph of Focus(Flat CDR) - these give you more CDR early game, which allows you to use your spells more often and last hit with your q more.
Greater Glyph of Celerity(CDR @ 18) - these give you more CDR mid/late game, which allows you to use your spells more often, giving you more kills.
Greater Quintessence of Insight(Magic Pen) - these give you more damage, cutting through some of their magic resist.
Greater Quintessence of Force(AP @ 18) - these give you more nuking capability late game, allowing you to get more kills.
Greater Quintessence of Focus(Flat CDR) - these give you more CDR early game, which allows you to last hit with your q more often.
Greater Quintessence of Celerity(CDR @ 18) - these give you more CDR mid/late game, which allows you to nuke more often, giving you more kills.
Greater Quintessence of Fortitude(Flat Health) - these give you some early health, giving you some more survivability

I use a common 9/0/21 Mage Build on Veigar.

With 9 points in the offense tree, I get some extra AP, CDR and Magic Penetration, which is great for any caster.
With 21 points in the utility tree, I gain some mp5, reduced cooldowns, movespeed, and a wonderful exp boost.

"Good Hands over Perseverence? LOLUBAD":
Perseverence, in my opinion, is useless compared to Good Hands. Once you get 100HP or MP5, you will gain additional 4 HP or MP5, with rank 3 of Perseverence. This is almost completely useless. With rank 3 of Good Hands, you are revived 6-7 seconds earlier than without it, which can change the outcome of a game from loss to win or win to loss.
"Why didn't you take Greed? Some of these items are expensive, and I want to get them earlier!"
Greed gives you 360 gold per hour. That's just 10 gold over Boots of Speed. In my opinion, it's just as useless as Perseverence. DO. NOT. TAKE.

Equilibrium - Passive - Veigar's mana regen is increased by .75% by each 1% of mana missing.
This passive is awesome. Combined with your seals and early tear, you should almost never go out of mana.
Baleful Strike - Q - Active: Unleashes dark energy at target enemy dealing 80-260 (+0.6 per ability power) magic damage. If the target is killed, Veigar gains 1 ability power.
Passive: Veigar gains 1-5 ability power when he kills a champion from any source of damage.
This is the reason many Veigar's are bad, and have nearly no AP. LAST HIT WITH THIS. I can't stress that enough. Once a minion gets to about 100-150 health or less, throw this at them. Congrats, you just gained 1 AP. You must last hit with this from the beginning to the end. Your AP will NEVER cap, and you will just become more strong as the game goes on.
Side note: toss this at an enemy with a Banshee's Veil. It will pop the passive on the item, and allow you to stun and get off your entire burst combo. By the time the stun wears and you've done your other abilites, this one will be back up.
Dark Matter - W - After 1.2 seconds, dark matter falls from the sky to the target location, dealing 120-320 (+1 per ability power) magic damage.
This is your nuke. Once it's maxed, you will clear out entire minion waves. Pair this with teleport on a turret being pushed by minions, and you'll easily gain 10-15 creep score.
Event Horizon - E - Veigar twists the edges of space around the target location for 3 seconds, stunning enemies who pass through the perimeter for 1.5-2.5 seconds.
This ability is AMAZING. It synchronizes insanely well with Dark Matter. An AoE stun? What's not to love?! It can turn the tide of any team fight in your favori. Just throw this down, then lay your Dark Matter. This combo will drop them, depending on their magic resist, from 100% to anywhere from 90-40% most often.
Primordial Burst - Ultimate - Blasts an enemy champion, dealing 250-500 (+1 per ability power) plus his target's AP in magic damage.
"Ohai Veigar! I'll just walk up to you an- *BOOM!* OWOWOWOWOWOWOWOWOWOW! *runs back to turret*" Welcome to Primordial Burst. This is Veigar's ultimate, and by far his most important spell. It allows you to nuke most champs from 50% to dead. Did I mention it stacks not only from your ability power, but also your target's ability power? This means any caster is completely screwed once you see them.

I most often use Flash and Teleport for summoner spells on Veigar.
I use these two spells because Flash is great for escaping, or catching up to nuke an enemy.
I use Teleport because, in my opinion, Veigar is one of the best champs to use as a make shift assassin. You can teleport onto a ward in the jungle and pop out of no where for a gank. It also gives you great map mobility, along with defending capability. Overall, it's a great spell to have on Veigar.

Other Viable Spells

Ignite: For when you aren't quite able to completely kill someone with your spell rotation, helping with an early kill, or when you have a pesky heal user.
Ghost: This spell is great. It can help you escape, chase, or get back to your lane fast after going to base. I usually take this instead of Flash, if I use it.
Exhaust: It can help you with going 1v1 against any champion. I almost never use it on Veigar, but when no one else on my team has it, I take it.
Cleanse: I've never taken this on Veigar, but it can help with removing CC and preventing a death.
Clairvoyance: I take this when no one else on my team takes it. Remember- vision wins games.

Non Viable Spells:

Heal: All other spells will benefit you more. You will often be called a noob for using this spell, and it's really only useful early game. Don't take it.
Clarity: When you rush an early tear, this item is almost completely useless for Veigar. Save it for the supports.
Revive: Not even going into it. It's pretty obvious.
Rally: See Revive.
Fortify: Leave it for the tanks/supports.
Smite: Are you a jungler? No. Therefor, you do not need smite.

The order you level up your abilities is VERY important for Veigar.
Q-E-Q-E-Q-R-W-W-W-W-R-Q-Q-E-E-R-E-E
With this ability sequence, you are able to farm early with your Q, as it does more damage. Your W becomes a huge nuke mid game, which is where it starts becoming a viable spell to use, as it costs a HUGE amount of mana. I max E last, as you do not need much of a stun to burst. I take a point in R whenever possible.

Your most common spell combo will be E-DFG-W-R or Q. By the time your ult hits, they will often be almost dead anyways. If they are, use your Q. If they aren't close to dying, use your R. With time, you'll learn how hard you hit throughout the game, and this will be easier to judge.

You can build Veigar in different ways, depending on the team you are facing/the teammates you have.

Start with a Sapphire Crystal and 2 Health Potions. This will give you some early mana to use spells and allow you to build an early tear.
On your first trip back, buy your Tear of the Goddess. Start getting your tear farm asap. Rushing a Tear will allow you to spam your spells.
After, buy your Boots of Speed, then an Amplifying Tome. If you have enough, buy a Kage's Lucky Pick. The Pick will be eventually built into a DeathFire Grasp.
Once you have your Pick, finish off your Boots of speed, and build them into Sorcerer's Shoes. This will give you some extra magic penetration, making your spells hit harder.
Build your Pick into a DeathFire Grasp. This is when you will really show through. Along with Zhonya's, bind this to a key you will remember. You NEED Deathfire Grasp to be a great Veigar. With an early DFG, you can shut down almost anyone with an E-DeathFire Grasp-R-W-Q combo.
After your DFG, build a Deathcap. This will make, not only make all your spells hit harder, but make your DFG's damage percent jump.
Build a Void Staff after Rabadon's Deathcap. This will give you more Magic Pen, but it will also allow you to crush them.
After Void Staff, finish your Archangel's Staff. This will make your AP skyrocket. By this time, the game is almost over. This item is the icing on the cake, in my opinion. It turns your already insane mana pool into a huge AP boost.
If the game is still going on, buy a Needlessly Large Rod, and build it into a Zhonya's Hourglass. Zhonya's, in my opinion, is a burst mage's best friend. The active on it allows you to wait for cooldowns and take no damage for a short period of time. I keybind this and always put it in item slot 6. Keybind it to a key you will remember, as you'll use it often. Zhonya's will also help with your squishiness, by giving you 50 Armor to help counter those annoying auto-attackers.
FINAL BUILD.

Mejai's Soulstealer: I take this item occasionally, if I'm doing extremely well within the first 5-10 minutes of the game. Get it between your Kage's Lucky Pick and Sorcerer's Shoes. Replace this with Zhonya's.

Abyssal Scepter: Get this item instead of Void Staff are constantly being destroyed by other casters, which shouldn't be happening. Also, you can take it sometimes if your team is very AP heavy, letting your team do more damage.

Rod of Ages: I used to take this item on Veigar, and I suggest it for newcomers to Veigar. It gives you some health, mana and AP. Instead of rushing a Tear, rush a Catalyst. Finish off this item first, instead of a Deathfire Grasp. Get your DFG next, and skip over the Archangel's later in the build.

Banshee's Veil: I get this item occasionally, to help with ults and stuff, like Karthus's, Ashe's, etc. The passive helps, along with the mana proccing your Archangel's Staff. Get this instead of a Zhonya's.

Thornmail: When your enemy is running AD Cleave, rush this. Get it between your Kage's and Sorcerer's Shoes. They will often target you, as you're extremely squishy. This will help your team a ton. Get this instead of Zhonya's.

Will of the Ancients: I get this item if I have a lot of AP on my team. The spell vamp helps everyone, and it gives your allies extra AP. I'd rather have a support get this, though, as it is mainly an aura item.

Morello's Evil Tome: I've never got this item, personally, but I have seen it used before. It's a great item, and gives you some amazing CDR, MP5 and a nice amount of AP. Take this instead of Void Staff if the opposing team isn't stacking Magic Resist, or Zhonya's if they aren't very AD heavy.

If I am missing any items you think would go here, leave me a comment and a reason why you think so!

Well well well; the guide finally comes to an end, and we part ways. Please comment on the guide, giving any suggesstions/feedback. Please do not vote down just because I do not play the same way as you do; we all play differently, and this is how I build/play Veigar.
Thank you everyone, and have fun!

PS: Didn't realize Cheatx had posted almost the EXACT SAME guide as this just a few days ago. I've used this build for awhile now, just givin' him a shout out to another good Veigar player :)