Anivia is a really overpowered hero if you can play him . It's really important to know the skill combo . So have fun with this guide and good luck! Sorry for my English , gonna improve it .

9x Greater Mark of Insight (to at least a bit magic penetration)
9x Greater Seal of Force (u dont buy much ability power in early game so with this seals you have a bit ability power)
6x Greater Glyph of Insight (to have more magic penetration)
3x Greater Glyph of Force (to have ability power again)
3x Greater Quintessence of Fortitude (to be not that squishy)

These are the normal masteries for a magic .
Offense: until you get magic penetration
Defense:/
Utility:if you use as a summoner flash take one point in Blink of an Eye , if you dont take flash take greed , the others are normal for a magic .

1.Sapphire Crystal + 2 Heal Potions , you need mana to stay in mid long enough .

2.Boots of Speed , nothing to say to it .

3.Catalyst the Protector , catalyst is good in early game because you can harass your enemies and you get life and mana back .

4.Amplifying Tome -> Mejai's Soulstealer , if you think you can stack it buy it , if you dont think so get Tears of the Goddness (5.) first and then Rod of Ages so you dont die fast and can resist the first skills, if you see that they never focus you , you can also get Will of the Ancients if you want, so you help your team and you get a bit spell vamp (good for ultimate) .

5.Sapphire Crystal -> Tears of the Goddness , so you have a lot of mana and you can do your ultimate without mana problems .

6.Boots of Swiftness , normal magics get sorcerer's shoes but anivia is really slow so she hasnt' got a good mobility . It's your choice but i play better with a good mobility .

7.Archangel's Staff , a perfect item for anivia you get lot of mana and ability power .

8.Catalyst the Protector -> Banshee's Veil , so you dont die fast against veigar for example , because if you die fast your ultimate is useless .

9.Needlessly Large Rod -> Rabadon's Deathcap , a must have for magics much ability power and
the 30% are just indispensable .

10.Needlessly Large Rod -> Zhonya's Hourglass , usefull because in the end game most dps are really strong (for example:Master Yi or Tryndamere), so you can do your skill combo and then activate it and wait for your cooldowns . The dps focus someone else and you can kill them . I think it's better than a health item because so you can't focused anymore for example if you get +630 health (Rod of Ages) you die with three hits from just one champion , a dps champion , (if more than one champion focus you , you have no chance) . But with Zhonya's hourglass you can't get focused from all and dont die .

11.If u cant stack mejai's sell it and sell boots of swiftness and get sorcerer shoes and lich bane.

I pick my q because in a early fight you will win with your stun and it does more dmg at the beginning than your e . After that i try to maximum my e to have save damage and if it's chilled you do lot of damage really fast . I just take my w on level 4 (later level:14,15,17,18) . If you can't pick your e , maximum your q . Of course on level 6,11 and 16 you have to take your ultimate .
If you combinate your skills you can do fast damage and get back then to wait for your cooldowns or you kill him .
Start with your r to slow , put the wall behind him to hold him into your ultimate, do your e to have maximum damage , use your q now (look on which side he wants to escape and stun him)

I use flash and ghost because then you can escape really fast because if you get focused u die fast and can't help your team . The two summoners are also good offensive summoners to follow the enemies . If you want you can take exhaust to have a save stun .

Last hitting is really important for anivia because the items are expensive . Until you get level 6 it's a bit difficult for some people to last hit but when you have your ultimate just do it to get the gold from all minions . I dont play anivia that aggressive because i think it's more important to farm a lot .

The blue buff is a must have for anivia with it . You can spam your ultimate without any mana problems .
The red buff is useless for anivia , give it to the dps .

Anivia is one of the best gankers in the game because she has a slow , a stun and the wall is really good for holding them in your range .
In team fights stay in the back of the team , do your skill combo and try not to get chased by your enemies . Your stun is really important in team fights so dont waste it .

First you go mid to get the most experience because you need to reach level 6 fast to farm good (7.Farming) . You can decide if you want to play aggressive or not , the egg is is really great in early game if you play aggressive . Because you attack the enemy and normally both have low life then . If the enemy stays now in mid play a bit more def and dont waste your mana after that let the enemy "kill" you next to your tower (not before your tower). You will reborn and maybe you get a kill then . Of course the other player has to be really bad . If you dont play aggressive farm as much as you can with your ultimate and do your other skills too because you dont use your mana for attacking . After a few time go back if you can get catalyst .

In the early middle game you dont have ability power (just your soulstealer and your runes) but you have a lot of mana and you are not really squishy . With your Glacial Storm you can easy get assists or kills for your soulstealer so you get ability power step by step . When you get archangel's staff you have something like 110-160 ap , it's enough for mid game because your ultimate does damage if the enemy stays in it and not fast damage so . The last item in the middle game is banshee's veil . With it you can survive the first wave of attacks and dont die (except everybody focus you) .

When you get Rabadon's Deathcap you have unbelievable much fun with anivia . She does a lot of damage now (Before she did also but you know with Rabadon's deathcap every magic does a lot of more damage ). Finally when you get Zhonya's hourglass you are really overpowered because how i told (4.Items) you dont die now to the enemy dps you just kill him easily . Now you are able to decide the game if you die fast in a team fight you will lose otherwise you will win . You have to foc us their dps and kill him as fast as you can . It's also your task to do a good wall so 2 champion are in front of it and 3 champions are behind it . Never forget to get the blue buff . I can't give you more tips how to play because every game is different .

I hope i could help you a bit . I tried to keep it short so you dont get tired of reading . Have fun with anivia .

Please vote and let comments , so i can improve my guide .