Hi, I'm DigitalMan. I'm not a Diamond or anything, but I know a lot about Amumu, so if that helps you win some games, great. I play him probably 70% of the time in my main role of jungler. (My wife plays support, so I get what I want a lot), and I'm level 7 with him. I've peaked at Silver until now, but life changes have been keeping me off the rift a fair amount, so I think my game knowledge goes much deeper than that rank would indicate.

Why Amumu? Because your team comp needs it. EVERY... FREAKIN.... TIME. He's not the champ your solo-hero teammates deserve, but he's the champ they need right now. Playing Amumu well is a matter of covering his weaknesses and optimizing what you can contribute to your teammates, but it's possible to carry on him as well, even with full tank build.

Pros:
Tank
Hard engage
Hard CC (decent ganks)
Disengage
Significant AOE AP dmg plus MR shred for your mid
Anti-Tank mechanic in your W
Strong late game

Cons:
weak early clear
Weak early game in general
Weak duelist
Mana hungry/blue-buff dependent
unreliable escape
relatively flash dependent for engage.

There are 3 main reasons to pick amumu

A) your teammates are picking AOE Damage/AOE CC champions
Champs with these type of abilities play well with Amumu's stun, role, and damage. It makes for a good teamfighting comp, and also opens the door to some serious 4-1 split pushing.
Examples include Kennen, Fiddlesticks, Miss Fortune, Sona, Rumble, Orianna, Syndra, Luz, Morgana, Zyra, Zilean and probably 47 more I'm not thinking of right now. You get the idea.

B) your teammates are picking champs with skillshots, especially AP ones
When you're holding someone still, it's hard to miss a skillshot. Ahri, Lux, and Morg come to mind.

C) To fill the holes in your comp.
This one comes up a lot, especially in a lethiality meta. You're 5th pick, and your team picked champs like Riven, Yasuo, Zed, Varus, etc. across the board. You have no tank, no AP, not enough CC, not enough initiate, and not enough peel. Picking Amumu gives you all of that. You're not necessarily going to play a front-to-back teamfight very often with this comp, but by rounding them out, you can open up more options for more plays, as opposed to picking another lethiality assassin.

Tank is 100% the way to build, 100% of the time. Why not build a couple of AP items?
You can try a Rumble-like build, but you'll see quickly that the damage you gain is HEAVILY outweighed by the survivability you trade off. The reason for that is that A) all of your moves put you within range of taking a beating, B) your moves are designed for taking damage and being next to the enemy for a while C) you have strong base AP stats and an MR shred anyway, moreso even than defense, so you're not gaining what you think you might be, and D) your W is partially by percentage health of your enemy, which obviously doesn't scale by AP. (only half of its damage does.) Finally, you picked Amumu for teamcomp reasons. You'll see in the champion match-up section, he's not usually a great duelist. If you don't build tank, you're not contributing what your team probably needs. Abyssal and Icebourne are more borderline bruiser items you can take, but if you're not building tank, you're doing it wrong.

Your first and highest priority is hitting level 6, so farm hard. Keep in mind that you do mostly AOE damage, so Gromp and Krugs are less efficient camps for you. Generally, I go Full clear, starting blue, to Gromp, and then straight through from there. I also recommend this route if you're new to Amumu, as he's very mana-hungry, and you will have a hard time managing his mana before you're familiar with his moves' cost. This season though, with raptors giving a bit of extra XP, you could start there, grab red, and clear the other direction. Advantages there would be that you have great AOE to help with that, and you aren't where your opponents expect you to be until they see you on a gank or something.
This would also be good against junglers that will try to catch you early. 98% of the time, your opponent thinks thus: "oh, I'm against Amumu. I better either A) kill him at blue or take his blue, or B) take his first red for free. That should take him out of the game while I gank 3 lanes for an easy W." But you can actually start red-side if you need to, which most people don't know. Better still, you take their red in trade, especially as their raptor camp will take you 2 seconds to be a camp ahead. It's better to take the trade and not fall behind than take a fight because A) you'll probably lose, you suck early on, and B) You'll be stronger later, so if nobody snowballs, you win.
Plan your subsequent routes so that you land on blue as soon as it's up, and pull it into bush to make it harder to steal it or gank you. Donating it, when possible, can make sure that it doesn't fall into enemy hands, too, because a 2v2 is more manageable for you.

The hardest thing about playing Amumu in a teamfight is not overdoing it. It's really easy for you to flash in, catch them all, and be upset at your team for not winning the fight. You need to make sure everyone is there and ready to go before you initiate, that they're in a stronger position and can win the fight, and most importantly, that you don't initiate so far out ahead of them that they can't get to you in time to win it.
It's also good to try to R people that will then try to get to your backline, right off the bat, moreso even than their carries. It gives your team free uptime, and the only person their team can hit is your tanky self. In other words, you start the fight with a huge positioning advantage that way, because instead of just hitting their tank, your team can probably delete their assassin or bruiser.
The other factor is your own cooldowns. Your team is even with their team. You're coming in from an angle, so you flash, Q, and R their team. Vi pops QSS and dumpsters your carry. What can you peel her with?
A mediocre Amumu player blows that Q up front. Sometimes you have to, but if possible, straight up flash/ult is much stronger. You can chain stun their carry with a Q if the situation allows (which can also let you get to their backline), or peel for your own team. Finally, one of the biggest knocks on Amumu is that he doesn't have an escape once he's in. That's because you're blowing your escape to get in. Q can be a way to get out, too, by snagging a camp or a minion, or even a farther away champ.
The same goes for ganking. Don't blow your Q to get into lane if you can just run up behind them. It's way more convincing to catch them with it as they're running back to tower after stunning you off, and it's the difference between a kill or a flash, and just wasting time.

Amumu is a delightfully simple champion that let's you focus on macroplay and supporting your teammates, as well as rounding out nearly every team comp. His ban rate is minuscule, and his pick rate is reasonably low, so he's very spam-able as well; definitely someone you can climb with. (and a good choice for beginning junglers)

I hope you enjoyed the guide, and win a couple more games because of it. I'm very open to criticism, so lay it on me; I can learn a lot from this too.