What is Anivia's role in a team?
Basically, Anivia is an AP Carry. Her nukes are awesome throughout the game and her passive is really useful in any phase of the game too.

Pros
- Passive helps in laning phase as Anivia doesn't have to go back to base for regen so much
- Awesome nuke
- Excellent harass skills
- Wall acts as a temporary sight
- Ulti pushes minion waves fast

Cons
- Extremely squishy
- Hard to master W
- Needs practise to get used to positioning wall and ulti
- W may harm your team if not used properly

Runes?
Red Magic Pens
Other runes can be a mixture of flat AP runes and AP at level 18 runes.

Masteries are pretty self-explanatory. Offensive masteries are standard AP carry masteries. In utility, however, I added the neutral buff masteries. This is because Anivia is quite dependant on the blue buff, so one may want to have it for a longer duration.

Focus on AP all the way! Anivia needs as much AP as possible to nuke. Of course some defensive items are needed, for example, Rylai's for health and Banshee's for Magic Resist and the amazing passive!

Why max Frostbite? It's because frostbite is Anivia's main nuking skill. Her Flash Frost + Frostbite deal fantastic damage throughout the game, thus she can easily take out squishies using the Q + E combo!

What can your W do? It can act as a temporary sight in an area, block escape/chase routes or even single out an enemy unit from his team! Timing and positioning W takes practice to perfect!

Take middle lane! Anivia is at her full potential at mid! In her first few levels, you have to last hit as many minions as you can. Anivia needs the gold to get items to nuke! From level 6 onwards, however, you can clear minion waves easily with the R + Q combo. By late game, you'll reach a really high AP amount if you farmed well!

When you're laning at mid, it's always good to know who you should play safe/aggresive against.
When you face Champions that can silence, do play passively and switch to aggresive play style when needed. Anivia relies on spamming skills to harass/farm, thus silencers can easily block out your skills for a few seconds.
Examples of "silencers" : LeBlanc, Kassadin, Malzahar, Talon, Cho'Gath etc.
Anivia also has difficulty with champions that can teleport a short distance because her Q may miss when the enemy teleports.
Examples : Kassadin, Ezreal, LeBlanc, etc.
Anivia lanes well against melee champions since she harasses alot when positioned right.
Examples of champs weak against Anivia at mid : Gangplank, Rumble, Pantheon, etc.

I hope this SHORT guide helped some of you out there! This is my first guide so I'm sorry if it isn't a decent guide. Do mind my poor English too since I'm an Asian :P
Well, see you around and do remember - Practise makes perfect! I can Garen-tee it!