Hey

Welcome to my first guide ever! I'll try to explain how I normally play Veigar and what makes him so OP. ^^

In my runepage I get 10 magic penetration (9x and 1x ), so you have 30 magic penetration with your . This way you can hit for 100% against all squishy champions with 30 magic resistance and perhaps force them to get magic resistance, or continiue pwning them untill they do or you win the game.
In my seals I go with 4x and 5x Greater Seal of Replenishment, so you can last hit more minions in the start, and it works well with your passive. I go half per level and half base, since you'll need more later in the game, because your ultimate costs a hell load of mana.
I take 9x in my glyphs, so you don't get bursted instantly from other AP carries or other Veigars. And I also use 2x , to make him more stable for early game.

I prefer and above everything else, as ignite is a nice addition to Veigar's burst, and it able to finish anything off, if you're out of offensive spells. is a must have on Veigar, because you can use it over walls to finish an opponent or get a nice on the other team or to flee, if you're getting hardly focussed and give your team some free time on the enemies.
Instead of , you could take , as it will save your life sometimes and can also give you a really nice 1v1 advantage throughout the laning phase.
can be useful as well, since you'll have 2 summoner's spells to increase your utility so you can either get away or chase an enemy easier.
is pretty awesome too, as it makes you gank other lanes faster, or can get you back into laning far faster than running, and you won't lose much expirience. But the downside with is, that it has a very long cooldown.

I always start out with a , as it will make laning and last-hitting much easier. And I also take a to stay in lane and a , so you can keep last-hitting untill you have to go back the first time.
When I go back the first time, I buy a , and if I have enough gold also a Kage's Lucky Pick. And some potions if I feel like it.
Then I go back when I have enough gold for my and (1350 gold), and then the fun begins.
After this it should be easy to get kills as you can one-burst all the squishy champions. Then the third time you go back you'll get all the AP you can. I always take as my next item, and always buy first.
Now you have all the AP you need for the mid-game, and now you focus on survive-ability. My favourite items are and , as they both give decent magic resistance/ armor and also some health.
And my last item is often a , to counter the tough guys.

Start out stunning the opponent with followed by shooting a above the place they got stunned. During the time the falls form the sky, you first throw your at the opponent, as it deals damage based on their current health, it will deal most damage if it's thrown first, followed by your and lastly your if you think you'll need it.

Use Smart Cast!
I always use smartcast with Veigar's , , and , so you only have to mouse-over the enemy and then press the buttons. This makes the combo much faster and you'll have time to flee, if you go in and stun more than one opponent with . But remember not to smart cast , as you then can't completely hit with the edge of the on the enemy you want to hit (If you do this, he can't away and he won't stand a chance at all).

Veigar is a solo-lane/ midlane champion.
The laning phase as Veigar is the most important part to get a succesful game.
Early game, you'll focus on last hitting minions with to slowly increase your ability power. You shouldn't go aggresive until you get your , unless your enemy forces you to. Getting a while the opponent is tanking the turret can lead to extreme harrasment or even a kill.
Once you get your and , you'll be able to finish of squishy champions with the burst combo stated above. This is where the fun and rage begin. From here on you'll prefer champions whenever your ultimate and is ready. Else you'll keep on last-hitting minions.
Late game, when you get your , the enemy fears you and will try to focus you down. You'll just have to keep your distance, and stun the other team and burst one of them and then flee. You have to do this untill you get some defensive items, so you actually survive the team-fights without being instantly killed.
End game, you have too much armor/magic resistance/hp from and for the enemy team to focus you and kill you, before you get your of, burst one of them and flee or if your teammates are there, you can chase them together.