Anivia is a very strong Ap Carry, I like this build cause it let you to be tanky and its more easier for you to go into teamfights and clear it. Its very important to stack your Mejais to have the ultimate domination over your enemies.
Anivia is a champ who is easily to farm with. If u want to be as good as possible with Anivia, you really need to farm

Its also good to build Frozen heart on Anivia, If u meets a team with much AD's and u want to use a lot of ur spells

I think Froggen also use a build like this, but u can play anivia in many ways.

Rod Of Ages is also an alternative item on Anivia if u get it between 12-18 minutes.

FlashFrost
Flash Frost is Anivias stun, it let u to easily stun enemies to nuke em down with your Frostbite, Frost bite can be reactivated and when it hits an enemy they get chilled, wich means that u can use ur Frostbite to make more damage

Crystallize
Crystallize is an amazing skill but that many doesn't understand. You want to use it to keep enemies out of teamfights or block their way when they are chasing you or your teammates.

Frostbite
Frostbite is Anivia's main skill that makes all the damage, You want to use this when they are "chilled" to make all the damage. Your enemies becomes chilled when they are in ur ulti or has been stunned by your Q.

Glacial Storm
Glacial Storm is an AoE(Area Of Effect) ulti that let's you to easily farm creeps. But remember do deactivate it as it takes a lot of ur mana per second.

I like going 21/0/9 with Anivia,9/0/21 i also a great masterie page. But i think all of the masteries combinations works.