Amumu rocks. That's all there is to it. He is a very durable tank if you don't completely suck, can jungle extremely well(one of the best, second to only Warwick, IMO.), and can put out monster damage when played correctly. Hopefully, I'll bring you one step closer to building him the correct way to do any of those things(especially the "tank" part of it.). So, take a seat, and I hope you enjoy my guide! :)

Prose(see what I did there?):
Very high amount of AoE presence
Absolutely demolishes opposing tanks
Rather high damage output for a tank
Very durable if done right
Team mates love seeing you in a team fight
Great Jungler
Has a skin that has a present for a head.(God, I love regifted Amumu.)

Cons:
Very mana hungry throughout the majority of the game.
Relatively weak until level 6
Often needs help if you want to start at blue buff at level one
The box on his head looks much less impressive in game... it's just wierd.

My rune setup is built for an all-purpose Amumu. You could both jungle or lane very well in it. It provides early armor if you enjoy jungling(which many people think jungle Amumu is the best way to play him, but I advise against it if you don't have access to third tier runes.) However, if you absolutely hate jungling, and never ever jungle, I would advise to swap out the Glyphs for flat c/d reduction. Also, many people think that mana runes are a necessity to playing Amumu. I disagree. I can lane perfectly fine without the mana runes, so long as I can manage to get blue buff early on. If you have a jungler(and it isn't you.) in your premade group, if you have one, then you probably want to replace the seals with your mana regen seal of choice. The marks and quints are non-negotiable. They are invaluable regardless of whether or not you jungle.

I really like this build. I love the reduced cooldown on Flash, and more importantly, the increased duration of neutral buffs. Blue buff is, without a doubt, the best thing in the game for Amumu. He really likes his c/d reduction, and more importantly, mana regen. This kid can just blow through mana if you aren't careful, and blue buff is a necessity if you can get it. Sorry for going off on that tangent; back to Masteries. Lots of tanks really like the standard 0/21/9 build, but Amumu can get by without it. I don't feel like the increased movement speed is all that necessary, because if you need it, you will either be chasing, which you have bandage toss and your ult for, or you will be escaping, which you have ghost for. Instead, I spent the 4 points in the increased health, giving you a better starting position in the game, which is particularly useful for jungling.

The items shown should by no means be used in all circumstances. That is only if facing a mixed team, with a relatively decent amount of CC, but isn't leaning either way for or against AD or AP damage. There are many situational items that can be used in replacement of certain things, such as the boots. If there is an overwhelming amount of melee damage going around, get ninja tabi. If there is a lot of magic damage, or the team isn't otherwise doing a ton of damage from melee, get Merc Treads. Lastly, if you are just demolishing the other team, and don't need to worry about CC or dodging, grab yourself some Boots of Swiftness, and smile as your team pummels them into the ground. As for the rest of the items, use whatever defenses fit the occasion(Frozen Heart makes for a very good anti-AD item; the 500 mana is extremely helpful in those circumstances.). However, don't drop Rylai's or Sunfire Cape unless it's absolutely necessary to stack magic resist; these items are VERY good on Amumu. Sunfire Cape stacks with Despair, and with Tantrum, makes for a very heavy dose of AOE damage. Rylai's is applied by Despair, and so the slow is always up, making it nigh impossible for the enemy to escape a gank.

Tantrum is the first ability you should max out. If you can get a point in it at any time, do so. It makes for very good damage reduction from early game creep aggro, and does very good damage, at a surprisingly long range. Also, it's cheap, and the mana cost doesn't go up as you level. Despair also does this, and you will notice that golem buff is less important as you get more mana. So, knowing this, what's the most logical choice for the second skill to max out? That's right! It's Despair!!! Despair is also one of my favorite skills in the game, for the simple fact that you kill things by crying on them. Then, lastly you max out bandage toss. You may be asking yourself, "but bandage toss does more damage per upgrade than tantrum! why don't you max that out first?" Well, kiddo, Bandage Toss also costs a LOT more than tantrum to use, only affects once target, and doesn't provide that early damage reduction that is so good at the start of the game. You only take that first rank of it before despair for the stun, and the wall-jumping capability. Of course, you also want to be leveling your ult whenever it's available.

Ghost- A necessity for Amumu, in my opinion. It allows me to chase, escape, or to catch up to a team fight. It makes it easier to initiate, because I can avoid minions with it. A very good all-round spell for Amumu.

Flash- This spell all but guarantees perfect Curses. You can flash up to a group of enemies in a team fight, pop your ult, practically putting them at your mercy if they were trying to escape. It can also be used to jump over walls to escape, or to jump out from the jungle to gank.

Smite- This is the spell that allows you to jungle. It not only rocks the socks off of Golem and Dragon, it can be used later game to beat back siege minions, or super minions. Another fun thing is to steal someone's dragon or blue buff with it. In fact, if you are going againt an enemy jungler who you know starts at blue buff, and you know they are alone, smite their golem and then finish them off right from the start of the game. Replace Flash with this if you are going to jungle.

Those are the only 3 spells I use, and are the only ones that I think should really ever be used on Amumu. However, if you don't have access to Flash yet, You could easily take Teleport or Exhaust, and I wouldn't be angry at you.

Note: In ranked games, I believe every team should have one copy of both Clairvoyace and Fortify. You are not the champion to take these things. However, if your team feels you must take Fortiy, you have one of 3 options: Take it and replace Flash with it, Jungle, or tell them to blow it out their ***es.

Revive- If you take this, I will eat your firstborn son.
Rally- And your parents.