Anivia is one of the most powerfull casters in LoL.
She doesn't have just a great nuking potential, but moreover she has a great survivability due to her passive.
In this guide I will go to explain my way of building and playing her.
Therefore you should know that my playstyle and build depends on many factors like my opponents team.

Greater Mark of Insight

Magic penetration is quite important for Anivia as it helps to push her damageoutput in early as well as in lategame. Cause of this effect you should get some of those.

Greater Seal of Replenishment

Greater Seal of Replenishment

Anivia is one of the most manadepndent champs in League of Legends. Therefore you should get some of those, mana refilling runes. This is important for your early laning-phase, as it allows you to stay longer in lane and maybe you can harrass the enemy to get an easy kill later on.

Greater Seal of Defense

Greater Seal of Defense

Armor is allways good in lategame, ddue u force the enemys to buy even more armorpen to do more than moderate damage.

Greater Glyph of Shielding

MRes helps u to little the amount of damage gained by ur laning opponent, as long as he is depending on magical damage.

Greater Quintessence of Fortitude

Health helps u to get more sustain in lane and survive some more ganks without having to backport.

Masteries are very important for every champion, don't forget that.
I prefer to get a good mixture of all categories.
As all the other times feel free to change things here, but I came to the conclusion that it's the best way to play with these masteries.

Now we come to the main part of all guides, the Item-build.
As it has the most significant influence on the power of your champion, you should choose them wisely.
I don't have a fixed build, because you would be unable to adjust your playstyle to the enemys' champions.
Therefore I play her in different ways.
I will explain some situations and the Itemchoices in below:

• Your opponent is a caster with heavy magic damage, like Veigar or Vladimir.
  In this situation i would start up with an Abyssel scepterabyssel scepter, to decrease there Damage on me and increase my damage on them. Next i would go for the standard sorcerers bootssorcerers boots and then the Rod of Ages or the Archangels staffArchangels staff.
• Your opponent is a squishy DPS like Ashe or Vayne.
  In this case, I would recommend to go for a Rod of Ages for great survivability and DMG-output, as you are able to stand in lane for a while with the passive of catalysator of the protectorcatalysator of the protector. If the opponent doesn't do much damage on you, you can also choose an Archangels staffArchangels staff to increase your mana regeneration, but in all cases you should get the blue golem buff for the same effect. This allows you to spam youre Glacial Storm for farming and youre Flash Frost to harass the enemy.
• Tanky champs like Mordekaiser
  To be effective against such an heavy enemy, I would recommend to start with Abyssel scepterabyssel scepter, too. Furthermore your next item should improve your magic penetration again. For this effect you can chose Void Staff, as it allows more ability power as well.

As you see it's depending on many things, how to build Anivia.

Flash Frost

Frost flash deals a moderate damage. But its' powerfull CC does fit the lag of additional damage. Furthermore you can hit twice with it. The only negative aspect is the slowness of the ball. Also it's a skill shot => hard to hit with

Crystallize

One of the best skills of the game. With this skill you can easily turn a 5v5 into a 3v5 or dispart one the carry from the whole team to kil hm and get away after. Moreover its a great skill to save your allies. Also you can escape with it by hinder them following you.

Frostbite

Frostbite is your main damaging skill. You shoul only use it in combination with a slow like Glacial Storm or Frost FlashFrost Flash. The reason for this is that it deals double damage when the enemy is frozen.

Glacial Storm

Your ultimate is great for farming. But its als usefull to lower the AS of the enemy DPS, who is endanger your teammates or to slow enemys running off. It's also a great combo with Frost FlashFrost Flash, as it slows the enemys and make the an easier target for your Q.

When you start to get your Wall is up to you. I prefer to skill it late, because of the awful damage in the early. But most players prefer to get it quite early. Sometimes I skill it quite early, for example im assisting a sidelane.

Flash

Flash is one of the best skills in my opinion, becaue it flashes you away of opponents, whom you can easily escape then. Also you can put your wall between you and them after flashing away. I prefer Flash, because ghost doesn't allow you to get out of the dangerous zone that fast.

Ignite

Ignite is great in early but looses its' effectivness in lategame. I use it to finish of enemys after zoning, like I'm lvl 3 and they are lvl 2. This allows me to get First blood in most games.
Other spells that I suggest working good on Anivia are Clarity and Teleport.
I wouldnt use any other spells, as they could be less effective.

I would suggest you, to do the blue buff at lvl 9 or 10, you should be able to do it, by using all your mana. After you got the blue buff you can spam Flash Frost to harrass the opponents' champion and to farm with youre Glacial Storm, therefore you should get the enemys blue buff while your one is running out.
Do not farm to much, just try to get last-hits in early and midgame. This allows the opponent to push to your tower and you can kill him, without letting him any chance to escape to his tower. After you killed his tower you can farm as much as you want, but be aware of misses on other lanes.

Zoning is a powerfull technique in LoL, therefore i would like to refer to this guide by schurelia: Play Video

Pros

• great nuker
• great farmer
• very usefull in teamfights coz' of the heavy cc & wall!
• great survivability in early due to her passive
• easy to escae with

Cons

• Slow
• many skillshots
• mana inefficent

All in all i hope this little guide about Anivia was usefull to you.
Moreover I want you to be forgiving about all my mistakes, because it is not my native spoken language. Also I want you to only comment about the content of this guide and not about the guide itself, as it is still under construction. (Images and visuel work will follow)
The last words that are left to say:
"Have fun by playing Anivia, good Luck and don't flame if you loose, it's only a game that should entertain us and not affect our real life." ;)