Anivia is a really hard to play champion, but if you know when to use your Q, then the fight, is won. If you have 1-2 kills, then you can easily kill and AD carry in 2 hits (Q,E, maybe ultimate, if you are lv. 6). Anivia is vulnerable with this build champions, who outranges anivia(eg. xerath) or champions, who can perform quick jumps rapidly (eg. yasuo or leblanc).
You can easily counter those champions who has no ranged AAs(eg. katarina). You can also counter champions who has a one-time jump(eg. katarina, master yi). Wait until they jumps onto you, then use your ultimate on yourself, wait until they arrives with the jump, then use your Q and E. They will take incredible damage after lv. 6 if you are exactly following this build.

Anivia is a champion who can give heavy damage to anyone in few secs, but she is from "paper", so she can take heavy damage too. That's why you don't need to buy to anivia boots. If someone wants to kill you, you can escape using your ultimate, your Q and after lv. 13 your W. Boots is an item wich blocks anivia from doing more damage, wich is the key of an AP carry. In team fights, let the others tank out the most of the damage, because you has high range, so you can help them from far away.

An early Tear of Goddness is neccesary because of the stacking mana, wich will be the key of the high AP in lategame. Your first item need to be Rod of Ages, because it gives you mana and AP, and also a little bit of health, wich can save your life versus tanky champions. Rod of ages was nerfed, so it doesn't gives you as many AP in early game. So your next item must be a rabadon, to have some AP. Farm with your ultimate to ensure the optimal gold amount and to let the tear of goddness stack. You may have about 500-600 stacks on it already, so it's time to buy archangel staff, to give you more AP. Your AP in this stage is about 500-600 (THAT'S REALLLY MUCH). The final must have AP item is void staff, because of your DOT (damage over time) ultimate. The final two items is really not neccesary, but personally I recommend to buy a rylai crystal scepter and a zhonya.

Anivia is a really good champion if you can play with her. Watch out for fizz, yasuo, xerath, leblanc and lux, wich are anivia's greatest counters. Fortunately, these champions are rarely picked (except yasuo, so ban it always from ranked games if you planning to play anivia mid).
Anivia's greatest advantage is that she can counter assasins, such as zed and katarina, wich are often played on mid.