Anivia is a champion with tons of damage, especially in early game, where she has one of the strongest burst of the game (just behind Leblanc's). It is a very interesting character as she have lots of cc ( 3slows, 1 stun, 1 wall) and tons of damage.
It will show you my way to play my favorite champion :)

As Anivia's early is strong, I suggest lots of AP for the runes to make sure you get First Blood.
AP Seals and Quintessences
MR Glyphs
Magic Penetration Marks

For the masteries, 21/0/9 is a good choice, like other ap carries

Start with

Sapphire Crystal and 2 Health Potion and a Stealth Ward

First back
Try to get Tear of the Goddess and Sorcerer's Shoes, and Stealth Ward if you can
Core
Archangel's Staff and Rod of Ages
Late Game
Zhonya's Hourglass, Rabadon's Deathcap, Void Staff, Elexir of Sorcery

Note:
If you play against an ad like Riven or Zed, it may be wiser to build Seeker's Armguard before Archangel's Staff

Flash and Ignite for summoners spells !

R>E>Q>W with 1 W at lvl 4

After you have your Q at lvl 1, always max your E as it deals tons of damage, with a W at lvl 4 to prevent ganks. Of course max your ult at lvl 6,11 and 16.

As I said, Anivia is really strong in early game, on your lane, focus and farm and when you reach level 2, try to stun with your Q and if you succeed, do your E, it makes tons of damage. If you can't stun your enemies, do your E after slowing your enemy with your ultimate
-For the passive, it's very nice in early game, if you manage to make your ennemy dive under tower. If you die away from your tower and your jungler, it's hopeless ! In teamfight, you can try to use your passive in a bush and hope ennemies will forget you for 6 seconds :)
-Try to always have at least 1 ward in 1 bush, and you can use your wall in the other bush if you fear the enemy jungler ( it grants vision ), that's why I choose W at lvl 4 :)
-Ask your jungler to give you as many blue as possible, so you can farm easily and harrass a lot.
-Your wall is much more efficient in the jungle !
-In late game, your wall gets bigger and is very important, don't forget to use it!
Try to harrass enemy and when your team engages, do your ultimate on the enemy team, focus the carry with Q and E ( poof, he's dead). Then E every enemy slowed
-Some allies like Nunu or Ashe can slow ennemies for you !

Try to get as much blue buff as possible when you reach lvl 6 for a permanent harrass.

Pros
-One of the strongest burst of the game
-Lots of CC
-Easy to push your lane when you reach lvl 6
-Few hard counters ( Fizz, Zed, Orianna..)


Cons
-Hard to last hit
-low mobility
-mana reliant