First of all i apologize for bad english!!!
Now to start with build..... why Anivia?
One of the greatest CC champs in lol(has stun and slows, also great dmg burst which makes her dominating in lane).
Here im gonna do 2 a bit different builds, one is with mpen runes and ap item, and the other is with hp runes and ap/mpen items.
Here are some short words that i use to ease my writing: cc- crowd control,mpen-magic penetration,ap-ability power,ad-attack damage,dmg-damage,aoe-area of effect,wall-crystallize, oom-out of mana,cd-cooldown,ff-flash frost,fow-fog of war.

Pros
-lots of slows,stun and most important wall(if you use it right you make 5v4/5v3/5v2 fights from 5v5 situations)
-Great burst dmg
-aoe slow and dmg
-great passive(spec. in early and mid game)
-one of the best creep killer/farmer
-if given chance can harras rly good

Cons
-oom (spec. early and mid game)
-cant escape easy
-squishy
-got many aim skills so its a bit hard to play
-slow movment speed

Build1: In this build im going for mpen runes so my item build can be full ap/mana(/mana reg.)
Mpen marks- I think all ap casters should use those runes.
Mana reg. seals-Well anivia goes oom rly fast so its a good choise.
CD reduction glyphs- you get 15% cd what is great in late game to harras with your frostbite.
Mpen quints- well in first build those quints so my item build is ap (more dmg but squishy).
Build2: In second build im going for a bit safer way to play anivia.
Mpen marks- well best for caster ap champ.
Mana reg. seals-For anivia's oom problem.
CD reducion glyphs- for harras with frostbite.
Hp quints- To play a bit safer at start, to be more secure against early ganks( like evelynn).

So they are 9 0 21 as any other ap caster champ.
In offense i get to mpen(what is normal for any ap caster).
Now in utility i go for improved clarity becose ill be playing in side lane(so i can get my teammate same amount of mana),improved flash for shorter cd,i could get more mana but im use to my mastery tree that way so if you think its better to fill up mana.... well feel free to do so.
You can also take one out from greed and one out from clarity to fill up that mana, if you are playing with ignite instead of clarity(but thats a bit harder becouse you will be oom more often).

First skill has to be FF becouse it does most dmg at the time and it can stun/slow the enemy.
Ater your ff as first ability you need frostbite(try to use when target is effected by ff to do double damage).
Pick wall early to help your teammate escape or to prevent an enemy from escaping(and it can also reveal bush and fow.
When you get to lvl 6 try combining your ultiy with frostbite so you can get more dmg.
Best combo would be ulti(so your enemy is slowed and cant dodge that easy), then your ff, then frostbite.

Build1: Becouse i use all mpen runes in this build and anivia is very slow champ, i pick boots of swiftness to make her fast enough so i can escape.
Catalyst is great item to get in early game(you get some mana and hp after you get lvl up).
Tear can provide better mana regen.
When you built rod of ages and arc. stuff you get nice combo of those 2(rod gives you hp, ap and mana which is great for arc's passive that gives you ap through mana).
Rabadons deathcap is a must have for any ap caster champ becouse it gives alot of ap + effect that gives you 30% more of your current ap.
Deathfire grasp is a great cd reduction item and also a great item to initiate a fight.
And finnaly zonya....well it gives ap and its best item to save you from abilites like Ks ulti,vlads ulti.kate's ulti ......
Build2: Now that i dont have mpen quints i need it from boots and void stuff(which also gives ap), also i im now a bit safer so i can buy tear of godess before catalyst so i can get my mana reg. quicker.
Rabadon's deathcap and zonya are must have in my build becose they provide a lot of ap and also keep you safe.Also rod of ages and arc. stuff are great combo to get more mana and more ap.

Before last anivia's nerf i use to take flash(best escape tool, very useful for anivia becouse of her wall).
Try to use it in jungle where there are narrow paths that can be easily blocked by your wall, and you can also use it while someone is turret diving to kill you( put your wall and wait for him to come to you then just flash over it).
And i use to take ignite which is best to get sure you will get the kill, but now im allways oom so i use clarity(considering i am a side lane, its very useful to me and my teammate.
You can also play with ghost instead of flash, and teleport instead of clarity.

Try to get kills in early game so you can get fed for mid game.
Dont bother farming creeps till lvl 6, becouse when you get to lvl 6 its just ulti and watching money appear.
If you are going for mana buff try to be full hp and mana cose its hard to get it (you wont be able to get it till lvl 8(ofc. if your teammates help you can).
Anivia is best in team fights. Your ulti is great to combine with other aoe ulties becouse enemies cant escape that easy. Also your wall can change team fights (try to get in team fights in jungle so it would be much more easy to wall off enemies).
In open areas try to use wall when chasing, or to stop enemies abilites(for esc. nunus or kate's ulti).
Try to save your stun also to prevent abilities and to chase.
Anivia is great pushe becouse she can farm whole wave of creeps, but she cant escape gank easily so be careful.
Anivi also has a big range so try to position your self far from battle.
In rank games try to help your mid carry get some kills so he can dominate late game.
If enemy team has tank fest anivia is not great champ choise but you can do your magic in early game and just use wall and stun in late.
You can also consider banshee vail(for ganks like eve's or ashe's), or guardian angel that gives you magic res. and it can save you from time to time.
Also if you are using snowball items guardian angle is great choise to keep your streak alive.

Well that is my anivia build.
Pls dont flame on it, if you have any comment or ideas that can improve this guide i would appreciate it.
Also i will upload some pics and videos ( like some gameplay tutorial, so the new player can see how does anivia work).
Thank you for checking my guid, hope it will help you to improve your anivia play :)