First of all, this is a "glass cannon" build for our mighty Martian. Veigar has one of the greatest damage outputs amongst all so we're going to maximize those nukes and play a bit on the lam in order to survive.

Pros:

-Scary amount of nukes.
-Area-effect stun.
-Great escape mechanism: Event Horizon + Flash

Cons:

-Squishy
-Low mobility

Reasoning for
Greater Quintessence of Magic Penetration and Greater Mark of Magic Penetration is quite simple. We want to make our spells damage the enemy at maximum efficiency. Sorcerer's Shoes combined with these set of runes will fully drop down the magic resist of any champion who has not farmed additional magic resist items.

As for
Greater Seal of Ability Power
I shall state why not mana regen but ability power. It's basically because you will not have mana problem with this build since we have clarity for early game and the Archangel's Staff for the mid/late game. So instead of mana regen a little bit boost to our AP is cool.

Finally Greater Glyph of Scaling Cooldown Reduction, I prefer my Cd reduction per level since Veigar is a late game killer. At lvl 13 the reduction will be equal to straight Cd reduction runes and in late game it will be much better.

1. Reasoning for given item sequence

is our starting item. The 100 hp it provides is great for staying longer in the lane and the extra 15 AP is quite helpful for succeeding the last hits with Baleful Strike in early game.

is self explanatory, we need the penetration.

game style of Veigar should include staying in a safe distance from battleground and killing enemies with the spell combos as all other glass cannon champions should do, so Mejai becomes a core item for Veigar. And we need to start working on our stacks as soon as possible so we can utilize the AP income rapidly.

the first Archangel's staff is for mana stability and a good boost on AP.

comes right after that since the new stats that this item provides is simply amazing. 155 AP is an uber boost alone and the %30 extra income is in cooperation with our previously stacked AP's.

x 2 more for the known best way of maximizing the AP through the late game.

2. Situational items & Alternatives

If the enemy team has some decent tanks and your team is having trouble countering them can be a great option (instead of one Archangel's staff).

Or if the mages & squishy DPS champions of the enemy team has started to farm magic resist than obviously will be a good alternative for one of the Archangel's staff.

1. is our "must" summoner spell. Once Veigar is successfully ganked there is no way he gets out alive. Hence Flash is your emergency exit door along with Event Horizon).

2. is often my second choice. Reason for that is the possible early kills that you can get. Since you will be farming AP with Baleful Strike all the time, you will usually have your mana low in early game. But when you reach lvl 6 you have your great combo to kill your opponent so clarity will be your extra fuel tank at that point. I get bunch of First Blood's and early kills with the help of clarity.

Alternatives

a. could be a nice alternative for clarity since Veigar is a slow and pretty immobile champion. You can go back on forth on your lane without losing XP and risking your tower with the help of teleport.

b. is also another good option to increase your low mobility and enhance the escape mechanism.

Above all, always and always try to get the mid. Veigar is all about farming AP and that is quite hard plus inefficient when you are on other lanes.

Gameplay of Veigar is divided in to two main parts: The Farm and The Ownage.

The Farm:

Early game is your farming phase. In this part of the game you mostly want to stay passive (in order not to get harassed), constantly farm AP with your Baleful Strike and always be ready for ganks. Do not waste your Flash unnecessarily, in fact if not extremely important try always to save it for your escape mechanism.

The Ownage:

Your ownage starts as soon as you reach to lvl 6 and obtain the killer combo. Which is :

1. Event Horizon
2. Dark Matter
3. Baleful Strike
4. Primordial Burst

However notice that this combo is not enough alone to kill your enemy directly from his/her full HP at lvl 6. You have low AP at those lvls so always try to shoot your enemy with a couple of Baleful Strike and drop their HP to at least 3/4 so that you guarantee your kill.

As you proceed with your consecutive kills on the mid lane if you also succeed avoiding ganks from other lanes you will quickly stack some AP on your Mejai and get your combo power boosted up to the point where it will take out a single target alone.

When you reach to max lvl Dark Matter and Event Horizon has almost same amount of cooldown. Try not to disorient their cooldowns, use them together. But of course there are certain situations that you should use these spells separately. Throw Dark Matter whenever you see a bunch of enemy champs staying close to each other and not being so mobile. Also you will use your Event Horizon as a part of your escape mechanism when ganked along with flash or you will use it to initiate a team combat where you have the opportunity to stun multiple enemy champs.

In big team fights of 3v3 or more, try to target squishy champions of the enemy team (especially AP champions for the best use of Primordial Burst). When your Event Horizon is on cooldown always keep yourself at a safe distance from the battleground and when it is ready, try to land it in a way that you can stun maximum amount of enemy champions. Then you can cast your killer combo on a single champion to kill it or you may split your spells to different enemies(this is the best option if they all have low HPs) to contribute your team or even get multiple kills.