Amumu capabilities are to tank, gank, initiate, stun, jungle and turn team fights.
I find with just decent team mates your first ganks will give early advantage on lanes and mid-late game your ultimate is one of the best game changer abilities.

This guide is intended to be brief albeit complete. Perfect for quick reference.
First build is standard jungle build, second is a jungle build which rushes into GP5 and the third is a laning Amumu.

This sections lacks on many jungling guides unfortunately. This section saves lots of headaches. Now on to topic.
Amumu is rune dependent. I heavily recommend Summoner level 20+. (not worth spend IP on lower tier runes) And get need as many as you can get of this runes:

• Armor Seals
• Flat HP Quints

For Masteries you will need one point in Awareness (allows leveling up after Blue Buff camp) and enough points in defensive tree for surviving the jungle path.

After you setup everything do a custom game and test if you can do the jungle route.

Summoner Spells

Smite - junglers should take it because allows faster jungling, secure and steal buffs.
Flash - excellent for defensive and offensive uses. Master it and use as

• escape mechamism by crossing walls or get precious distance from enemies
• last resort correction of position to ultimate or surprise ambush on enemies
• flash out of team fights when your job is done and you feel you can survive
• counter enemy flashes or blinks (if worth it) and grab them with bandage toss

Viable alternatives:
Ghost if you prefer over Flash you are free to use it (or you do not have it or suck flashing)
For laning Amumu, replace Smite with
Teleport I see that makes you get quick to action and defend towers
Fortify for a premade team I comprehend if you take it
Clairvoyance if no support take it... wards can overcome lack of this
Heal if you are a beginner or do not have other Summoner Spells...
Any other is not recommended.

Masteries

The point in Smite. Amumu jungle is fast so the point in Smite does get use.
Get those defensive masteries for sustaining jungle and extra tankiness (avoid dodge as is counter productive to Tantrum).
In Utility, I spend points in Good Hands because you will die for you team. Awareness is a must for not staying behind in levels. Early you are dependent on blue buff so you need the buff duration. Greedy is optional.

Runes

Greater Mark of Magic Penetration x9 - give you more consistent magic damage all game long
Greater Seal of Armor x9 - reduced damage from monsters (common in junglers) and adds tankiness
Greater Glyph of Magic Resistx9 or Greater Glyph of Scaling Magic Resistx9 - for tanking against casters
Greater Quintessence of Health x3 - flat HP in order to make jungling possible and safer

For in depth description of abilities read:
http://leagueoflegends.wikia.com/wiki/Amumu_the_Sad_Mummy/Ability_Details

Cursed Touch -> make sure you auto attack your victims

Bandage Toss a skill-shot that throws you to the enemy and stun him, use it to gank, save allies, stop DPSers, chase fleeing enemies, jukes like toss into neutral monsters or minions or even enemy champions closer to safety. Master it!

Despair first you use it to jungle. Later on it will deal consistent damage in AOE that even tanks fear because it depends on % of their HP.

Tantrum nice passive that helps you through minions waves and jungle. The active accelerates your jungle, good farming tool and with combination to Bandage Toss makes you a better chaser or even harasser in laning.

Curse of the Sad Mummy game breaking ultimate. It snares all enemy champions in a generous area. Deadly when combined with other champions AOE abilities. It has long cooldown so use it when it is decisive. It is mandatory your team mates are well positioned (close, together and defined fronts) and if they are aware that you have ultimate and will initiate better. But you can use to secure a kill on 1 fed opponent as last resort. Rarely or even never use to allow escapes (not that 90% of players when see you use ultimate will not keeping fleeing).

Skill Sequence

Despair is needed for jungle, then Tantrum gets priority for faster jungling and more damage early on. There is no problem at all in puting one point in Bandage Toss at level 3 and gank. Bandage Toss gives the exact same stun time for all levels, the difference is cooldown, so sometimes I feel the CDR in Bandage Toss worth it (my team mates deal insane AOE damage and the stun allows to catch and kill remaining). Put points in ultimate at first chance.

Combos

Easy to figure out.
Gank: Q E W (auto attack) chase W again when done
Harass: Q E W W back
Initiate fights: Q (and/or flash if worth it) E W R (when positioned to trap as many enemies as possible)

Opening Items

Cloth Armor + 5x Health Potion - common in junglers and more sustained
Regrowth Pendant + Health Potion - rush into Philosophers' Stone, great for laning but in the jungle does no forgive if you take extra damage
Mana Manipulator if you go with someone premade and mana hungry (builds into Soul Shroud)

First recalls

Build your Philosopher's Stone on first recall/chance and maybe some Health Potion.
Or Boots if you feel need for speed.
Buy Sight Ward regularly and use them wisely. (they pay themselves if you get 2 or more assists from ganks that arise from map control!!!)
Buy Heart of Gold asap for extra HP that pays itself.

I have to test if Catalyst the Protector works great while jungling/ganking lanes and builds into Banshee's Veil. If you try leave a comment.

Boots

By around 19:00 you must have better boots (see if others have), so there are the options:
Mercury's Treads - against teams with CCs, you will need CC reduction to use your ultimate! MR is always welcome.
Boots of Swiftness - make use of superior speed to initiate, chase and rescue (very late game switch to Mercury's Treads)
Ninja Tabi - counter to strong AD team (against Tantrum but if they have like 4 AD you will still spam it a lot)

Core Build

Finally Buy Aegis of the Legion for tanking, help your team survive and deal more damage. Your core build is now finished:
Mercury's Treads Philosopher's Stone Heart of Gold Aegis of the Legion

Start to improve and adapt your tank prowess with some of this:
Giant's Belt if their damage is balanced or you need more survivability
Negatron Cloak / Null-Magic Mantle if their AP is fed or team is mostly AP
Chain Vest / Cloth Armor if their AD champs are fed or team with lots of AD
And you will build them for some of the options.

Options

Try to keep your Heart of Gold and Philosopher's Stone as much as time as possible. I sell Philosopher's Stone last.
Generally I buy Sunfire Aegis then game is over 80% the times. Next items are luxury...

Building for Tank
Aegis of the Legion core build, see above
Sunfire Aegis great on Amumu, as you already Despair on enemies, HP and armor are welcome
Frozen Heart counters AS and AD, and you gain CDR
Thornmail best counter versus heavy AD teams
Randuin's Omen uses your Heart of Gold into a great AD counter active
Banshee's Veil excellent counter to casters
Force of Nature good choice against AP heavy team, with good tank stats

Eleisa's Miracle if you did not get tenacity from boots you can replace with this...
Quicksilver Sash MR and breaks all cc for letting you ultimate
Guardian Angel mid-late game if all your team survives except you get this...

Warmog's Armor never used this on Amumu but you have already many MR and Armor get this
Leviathan stay away from this

Building Offensive
Abyssal Mask MR for you and enemies lowered MR, great if your casters are doing great
Rylai's Crystal Scepter your abilites slow and more HP, great if you won last 2 team fights and some fled with success
Will of the Ancients nice one if your casters are doing great and they do not buy it
Zhonya's Hourglass armor and a nice active, I can understand its use but not ideal

Building for Support (auras)
Aegis of the Legion see above
Shurelya's Battlesong nice stats for a tank, if you dont have a support and the active is useful plus it is a great use for your Philosopher's Stone
Soul Shroud may fit on some premade teams (like one with Karthus *tip*)
Will of the Ancients see above
Abyssal Mask see above

Note: do not forget to use actives. (press 1-6 keys above QWERTY) If you forget do not stop buying them, it is a matter of practise.

Jungling is about kill neutral monsters between lanes for experience and money, allowing one more of your allies to gain more exp and money, secure buffs, ward jungle and river sides of lanes as well and outside of the jungle gank, make pressure on enemy team, allow a team mate to recall while you defend his tower.

Amumu is very restricted to blue buff because of mana dependency. And can complete the entire jungle path bellow 4:00.

Faster Route: "Classic Blue Golem"

The camps sequence is: Blue Golem > Wolf Camp > Wraiths > [Red Golem] > Double Golems
Watch how to do it in this video: http://www.youtube.com/watch?v=f1AexsZbKlM

Alternative Routes

Can be used sometimes
http://www.youtube.com/watch?v=wYx8b8J5wHw

Leash or pull

Leash or pull means one ally will auto attack or some ability (ranged preferably) the golem so you more easily do your jungle and they go to their lane quick for not stealing experience. In Amumu case with all runes you wont need leash but it will allow/help to clear all jungle without recalling.
On game start ask for leash and protect jungle. Generally mid laner does the leash.

Ganking

This is the most difficult and most important task of jungler. You need to be aware of how your lanes are going. Killing neutral monsters becomes a easy task (except first blue kill) so you can screen on the lanes. If there is an opportunity unlock in level 3 Bandage Toss, pop Health Potion by the way and go for it.
Generally do not gank if your team mate is less than 1/3 of HP. It is too risky and both may get killed. And to guarantee successful ganks the lane should be pushed to your tower or to middle of it or the enemy is very low HP and you can tower dive (care with it).
Your approach is very important. You can get them by behind or from the side brushes. *videos to come* You need to mind the minions waves because they are a wall to your Bandage Toss.

Counter-Jungled

The enemy team sees your Smite and decides to invade your blue. Unless you are in low ELO, ask your team to protect the jungle and you by the bushes. Most times they will back off after see your team mates defending but a skirmish can happen and first blood. You can even trap 2 enemies if you position for ambush. *picture to come*
In the first situation wait to see in mini map all champions on their lanes. If they have a jungler, ask clairvoyance to check or just start jungle but religiously secure buff with Smite.
In the later, of course, help your team mates but try to avoid getting hit and try to spare your mana. Sometimes you will a strong leash or recall. On worst case do a recover path if they stole blue buff or you were killed and blue buff gets too strong for you (because his HP scales with champions level!).
Sometimes they enter the other side of jungle and steal a Wraith or Golem. Well continue your task but use wards and play smart.
Sometimes they invade the jungle to get you, use flash and bandage to flee until your allies arrive.

Counter-Jungling

Amumu is not a good counter jungler. Amumu does not has recover HP abilities or speed steroids, you can do it but you will need knowlegde of their jungler path, good team backup and excellent map/game awareness. Sight Ward and Clairvoyance is strongly recommended.

Warding

A jungler wanders a lot so warding comes naturally, here are important spots:

• Brush near the river in side lanes
• Dragon around 4 minutes
• Your buffs (specially blue)
• mid lane side brushes (when teams converge a lot in middle)
• their jungle buffs or crossways and gank their jungler (for premades or medium/high elo)

Even if you have a dedicated support ally, you should keep ward at a lesser extent because you have more income than the support and he can invest on Oracle's Elixir and finish more quickly his/her build to help the team.
*image to come*

Other tasks

An ally need to recall? Go to his lane, just last hit, avoid taking damage and when he is arriving leave immediately for not taking experience from your ally.

There is a skirmish nearby and you have HP? Go there and do something if you can afford. Rescue someone by stunning their chaser, even better if you can help kill or unfortunately you go clean up and you get the kills.

One lane got owned? Try to save the tower and do not let EXP and money from minions get wasted.

If you killed or forced enemies on bot lane to retreat try to get the tower and/or dragon. Amumu is great dragon killer because of his Despair. I recommed 1 ally assist you. You can solo dragon in level 10 or so. Your Despair and Sunfire Aegis speeds up Baron killing a lot so be there.

Lastly, if you are not getting killed by team fights you can buy Oracles' Elixir to end their map control or counter champions with stealth. ( Twitch, Evelynn, Shaco, Akali...)

Playstyle

You are not going to kill lots of enemies. You are going to help your team mates do that all game long. Of course, if you are in better situation (position and HP) to kill someone safely while your partner does not do it. Look at your score as K+A/D.
You save partners by stunning enemies with bandages tosses. Give priority to carries (most fed). Or stun with bandage their carries.

If you are jungling, make them fear your presence and as well your absence. If you are laning, make them fear your harass combo.

Take the lead of the team and be in the front. There is no general rule, you have to do the right choices from your experience and thought of the game.
Manage your mana wisely, mid-late game you have enough mana regeneration and pool, trust me. Early game ganks you may want to keep 180 mana reserve for Bandage Toss plus Curse of the Mummy. Without mana you are severely handicapped but auto attack for good use of your passive.

Early Game

If you jungle, make your ganks give you upper hand of game.
If you lane, do what is possible to assist your partner in a kill. Do not get freely harassed. Use brush to your advantage and Bandage Toss from there.
Never tried but keep your mind open to gank mid or invade jungle if you know what you are doing.

Mid Game

Teams converge to fight in 5vs5, you should be present in 95% of them and initiate or enter properly to win. Sacrifice yourself for your carries if you have too.
It is ok now for you to push and farm some lanes if no one else cans or you are way too close.

Late Game

You are always on front of group and lead the team. Combine your ultimate with allies AOE ultimates. Do not be afraid to die if your team kill them.
Split for pushing if you know what you are doing. Learn to retreat nicely after pushing enough. Warding river and jungle cross ways helps.
Ward dragon/baron all the time. Baron can turn games.

Team Fights

If you are way too ahead it can be as simple as bandage on one, ultimate on all and ace!
Ideally toss yourself to their AD carry or fed AP carry. If someone (not the tank) gets too in front of their team you can make them pay for that. ;)
If in the last 2 initiations or so you lost (3 kills for your team and 5 for them is a lost team fight), poke them enough, gank one that is alone for 4vs5 or wait for some kind of clear advantage for your initiation be worth it.

You do not have to engage on every single opportunity. Do not fall into trap in brushes. Learn to retreat when you cannot win or are in disadvantage. Do not wait without knowing what to do because you have to be decisive but the general rule is not doing stupid mistakes.
If you beat them all great. If some are escaping you have to ponder chasing or staying with allies because you are holding support items.
If did not go well and you are still alive you can help to retreat and save some comrades.
If you ended killed but all your team mates survived, excellent too. Sometimes I barely survived because of a flash but obtained an ace. Push AND/or get baron if you ace and retreat asap when they appear in enough force. (Tab to see death timers)
Oh if you build correctly you will notice that you do not get dead easily. Amumu is not the biggest HP/Armor stacker but holds a lot of pain.

Solo Queue Tip1: talk to your team and give some tips. Warn when you have ultimate. You can tell them you are going to initiate when you ping twice.
Solo Queue Tip2: if your team mates are generally too dumb try to be invite the best you find for matches until you get good teams or play carries until you get out of ELO Hell.

Amumu must be feared and tank for his team. Gank to give kills to your team. Chase and rescue with bandage. Ward whenever you can. Initiate all team fights in advantage or equal chance but turn it with your ultimate. Enjoy see your team mates do a ace.

About the guide, there is some clean up I would like to do. Redo some text parts, maybe put on lists and suggest more videos/images. Also make more new players friendly.