Simple but Effective

Reasoning:

This is a rather straight forward and reliable Anivia build. I main this champion and I have a huge success rate with her. This build is designed to be in a side lane supporting/babysitting another champion. The way her skills are designed she can be extremely aggressive early in the game and with the mana regen runes and the Doran's Ring you can typically generate enough mana to support constant harass (remember auto attacks are brilliant for harassing).

Summoner Skills:

Exhaust: I take this for its utility. Not only can it save your hide when you get caught alone or focused in a team fight but you can also use it to help shut down the opponent physical carry during team fights (and due to the improved exhaust taken in the masteries you can increase the damage that the target takes by lowering their Magic Resist and Armor).

Teleport: I take this for the utility as well. You have near complete map control by teleporting into team fights, tower pushes or just to defend a tower. My philosophy on summoner skills is whatever gives you the most utility will better aid the entire team. Which is why I don't go for skills like Ignite or Ghost because they are selfish skills and don't have any place in aiding your team (I would take ignite with champions like Malzahar, Kassadin and Mordekaiser because they can typically directly benefit from its use).

Masteries:

I go for a standard 9/0/21 for my casters. I go into Offense for Improved Exhaust and Archaic Knowledge, picking up the cooldown reduction on my way there. In Utility I make sure to get all the mana/mana regen I can while remembering to take Improved Teleport. I grab extended monster buffs, Greed, increased movement speed, more cooldown reduction and topping it off with reduced recharge time on my summoner skills.

Skills:

Your main source of burst is in Frost Bite. Setting up for Frost Bite is rather easy in all respects due to Flash Frost's large hit box and Glacial Storm only needing to tap your target to Chill them. In some situations I take one more point in Flash Frost early instead of picking up Crystallize but that honestly depends on if you actually need that little bit of extra damage. I mush prefer the utility you get early game by getting an early point in Crystallize. The basic skill combo goes as Glacial Storm -> Frost Bite; this will get that initial burst out there and typically make your target **** themselves resulting in them leaving the fight or attempting to run right off the bat. The more complex combo goes as Glacial Storm -> Frost Bite -> Flash Frost -> Frost Bite, somewhere in there you can throw out a Crystallize to complicate their pathing which will make them run through your Glacial Storm even longer and even give you a chance at that second or even a third Frost Bite if they aren't dead yet. Alternatively you can open up with Flash Frost for the stun -> Frost Bite -> Glacial Storm and then whatever works after that. Typically you need to be throwing walls around all the time to complicate fights for the other team.

Another thing to take note on is that both of her major damaging skills (Flash Frost and Frost Bite) have a 1-1 AP ratio. Meaning for every one point of AP you have those skills do one more point of damage. But there is a catch; you need to skillfully use these abilities to get the 1-1 ratio. Flash Frost has the potential to hit twice, the initial damage from the orb traveling over your target and then the explosion. If you can manage to detonate the orb right after it passes over a target then you can hit them twice (achieving the 1-1 AP ratio on the skill) and then you even get the stun which is just icing on the cake. Frost Bite is easy to double because all it needs is for your target to be Chilled. This only requires you to have already hit with Flash Frost or Glacial Storm.

Items:

I go for a simple and safe build that is hard to topple because it is so reliable. It may not give you the insane amounts of AP that a Mejai's Soulstealer build but there is a time and place to use it. I start with a Doran's Ring which will give you decent mana regen and a nice boost to your health. After that I rush a Tear of the Goddess and leave it at that for a short while. Tear allows you to build up a nice early pool of mana before your skills begin to eat up at your mana reserves. After that I grab a Catalyst the Protector in preparation for upgrading into Rod of Ages but before doing so I get the basic Boots of Speed. Once I have completed my Rod of Ages I finish the Tear of the Goddess into an Archangel's Staff, depending on if you need the early Magic Pen or more movement speed I sometimes get my Sorcerers Boots before upgrading my Tear. At this point you are fairly powerful and a force to be reckoned with, but also at this point you have to make a decision on where to take your build. If you are having issues with the other team stacking Magic Resist then go for a Void Staff for the Magic Pen. If the other team isn't really stacking any Magic Resist than go right on ahead and grab a Zhonyas Ring to simply overpower them. Either way you will end up with both of these items, it's just a matter of which one you think you need first. If things are going badly and you are having issues staying alive then you should probably pick up the Guardian Angel before both of those items.

Play Style:

Remember that you aren't focusing on being the carry; you are there to support your lane partner and make the enemy team melt in team fights. If you happen to get enough kills early game you will eventually turn into the carry due to you simply out-gearing the other team and getting AP faster than they can get Magic Resist. Anivia is one of the best farmers in the game. All you need to do is lay down Glacial Storm on a wave of Creep and send out a Flash Frost, the Creep will fold like paper. If you do turn into the carry then nothing much changes for the build because it is designed to counter everything the other team has to throw at you. If things are doing tremendously well then go right on ahead and pick up a Mejai's Soulstealer if you really want to (or if you feel that you will be needing the ridiculous AP later in the game, usually due to there being too many super beefy dps or tanks on the other team). Remember to always try and get the double hit with Flash Frost for the most optimum damage output and practice using your Crystallize often. Crystallize can screw over any attempt the enemy team will make at having a team fights or shut down any kind of chase they could go for. Anivia is all about controlling situations and punishing the enemy team for initiating on your team. If played right I firmly believe that Anivia is without a doubt the best champion in the game.

This was my first guide so please comment. I will take into consideration everything you have to say and edit my build accordingly. Thank you for reading, GL HF.