1. Amumu Jungle tank for 5's - The basic build. Use it when you are playing an average game. You will do decent damage because of your base and abyssal scepter and be basically unkillable.
2. Lane Amumu for 3's - In my opinion Amumu is the best tank in 3v3s because of his high damage and good cc. You can use this for laning in 5v5s but it is not as good as jungling. (Switch out Lucidity boots for Merc treads)
3. Amumu Jungle Ap tank for 5s - Use this build if your fed once you buy your sunfire cape. If you are fed you can burst their squishies from full to zero.

Pro's

Extremely hard to kill
Best damage for a Tank
Some good AP ratio's (A little AP goes a long way)
Great Juker because of Bandage Toss
Amazing CC

Cons

Mana Hungry
Minions Hurt his initiation abilities
Not as unkillable as other tanks

Amazing
Smite - If your jungling
Flash - Escaping, Juking, Positioning for your ult or Bandage toss
Exaust - If your laning

Ok
Ghost - If you suck with flash
Exaust - If your jungling your going to want an escape or chasing mechanism
Heal - If your jungling at a low level this will help
Teleport - If your solo top this will help in 5s but I would still go with Flash/Exaust (Switch it out with Exaust)

Bad
Basically everything else

These are the mandatory Masteries for Jungling Amumu. But for laning Amumu your going to want to go in utility for a stronger late game but a weaker early game you can.

The skill sequence in this guide is average for Amumu because if you are being hit by any autoattacks Tantrum will be doing the most dps of any of your abilities because of its low cd with despair doing the least no matter what. In a 3v3 if all of your opponents are AP and they DO NOT auto attack you then you can max bandage toss first for max dps.

3v3 Amumu
Using the AP build in 3v3 is completely fine if you are looking for more damage in team fights especially since towers do less damage in 3v3s.
Archangels Staff - A good switch with Guardian Angel if you are not good at conserving mana

Zhonya's Hourglass - Switch with Randuins Omen if you dont need as much Armor or the slow is not needed in teamfights

Mercury Treads - If 2 or more of them have stuns

Thornmail - If they are lacking Magic Damage Switch with FoN

Banshees Veil - If they have weak or no physical damage switch this with Randuins or Gaurdian Angel

5v5

Amumu

Thornmail - Look above

Banshees Veil - If they have few spells that they can waste to break Banshees (Switch with FoN) or if they have weak Physical Damage (Switch with Randuins Omen)

Boots of Lucidity - If they have 2 or less stuns/snares/Fears/Taunts

Aegis of the Legion - I am putting this here to put possible items you can switch it out if you get past your full build (Which is rare)
AP - Lichbane or Zhonya's Hourglass
Tank - Zhonya's Hourglass or Rabadons Deathcap

Amumu is regularly banned in ranked so here is how to get him. (dont try this unless your team is first pick). When it comes to your two bans ban to regularly banned champs that you (or an ally on your team) knows how to play well. At this time they have a choice to either ban 1 of the two champs and give you the other. Or ban neither
and get the one you dont choose for themselves. (Though the second only happens rarely)

With full runes at level 30
Start with cloth armor and 5 hp pots but DO NOT GET A RANK IN ANY SKILL. Go to your blue and wait with your team there. If the enemy team tries to counter jungle get tantrum if its a 5v5 otherwise despair is decent. If your team will win the fight go in and kill them. If they wont start at Wraiths-->Wolves--Double Golemns-->Blue-->Wolves. Then work your way down to double golemns
If they dont counter jungle or you win the teamfight with despair go to blue (Try to heal a bit before starting it if you were in a teamfight) have your mid leash and just pop despair with a hp pot and take it--> wolves (with a hp pot if you were a teamfight)--> Wraiths hp pot --> Red hp pot--> Double golemns (Hp pot if at less than 4 bars)

Yes there is a section in here for when you should give up blue buff

You must give it away

Nidalee
Lux
Taric
Oriana

If they demand it

Ryze
Alistair (But if you have two tanks you have a problem)
Morgana
Fiddlesticks (Early Game)

If you have a Soraka in skype/ventrillo who is willing to give you mana you can give your Blue to a AP carry that isnt capped on CDR

As Amumu you have a couple different ways to initiate a team fight but once you get into a team fight.
The Bandage toss method - The most know way for Amumu to iniate. Bandage toss in (preferably on one of their squishies) and ult once the rest of your team gets into the teamfight.
Bush iniation - Have your whole team Hide right next to you in the edge of a bush that the enemy team will enter. Once their whole team appears in your sight use flash or bandage toss to position yourself so that your Ult can hit every person on their team with one of their squishy carries withing focus range of your team.
Run in and Distract them (for AP build only)- You may take a bit more damage before the fight happens but if you cant run in and FOCUS their AD carry with your spells you can burst them down from full to 0.

Once your in a teamfight you need to stick to the players with the highest damage and spam your spells on them. NEVER CHASE WITHOUT YOUR TEAM. If an enemy starts to run and the rest of their team stays in the fight your team will die without you.

Because Plainly FoN gives you more MR and there are many spells on both supports, tanks, and AD carries that can be wasted to pop your spell sheild. There may be some hp on Banshees but both the hp regen (which helps some during fights and lets you push after fights) and the extra MR gives you overall more surviveability