This was my first guide - which is now slowly but steady is being turned into a jungling guide.
It isn't ready yet, but hope you'll enjoy the parts that is done...

Pros:
Very tough late game
Has two types of CC (BT/CotSM)
Pretty good AE damage
High survivability
Good Initiator

Cons:
One CC is a skillshot
Low mobility

I've listed some runes in the top, they are tier 3, but you should be able to get the picture.

Per lvl -Armor: because a little extra armor wont hurt, and you'll be a mean tank late-game
Per lvl -Magic Resist: same as above.
Per lvl -Magic Penetration: Cause most of the damage amumu does is magic (ie. Despair/CotSM)
Flat - Health: I went with flat because +78 health is a lot in the level range 1-7, whereas +100something health from three tier 3 health/lvl only awards that amount at lvl 10+, where 100 or so health isn't a big deal.

The above explanations also goes for lower tiers of runes. A lot of people will tell you not to buy runes, but all fresh and wet behind my ears i find them helpful, especially early in the game.

I went with 0/21/9, as Amumu really does his best lategame work as a tank.

The point in Summoner's Resolve is there for the improvement to Smite - duh =)
The point in Bladed Armor is there to deal some extra damage to minions early on..
Rest of the points in the defense mastery tree is there to reduce incoming damage, and highten Armor, MR and Health.

Summoner's Insight is taken to reduce the cooldown on flash.
Expanded Mind is simply to increase mana at the early levels. If you think you're gonna die a lot, or if you are new to jungling, the points might be better spent in Good Hands
Runic Affinity is a must, to increase monster buffs, the blue buff gets you a long way, especially first off.

Optional Items

This item is good for fight against high CC/DoT teams. If you find yourself not liking it, or if the enemy team doesnt have DoT's or CC's, you might wanna change this to another item with bonus to magic resist.


I like this due to the fact that it gives armor, CDR and debuff to nearby enemy champions attackspeed.


If you don't want Frozen Heart this is a very viable alternative. Active slow to nearby enemy champs, and a bonus to armor speaks for itself.


Again an item i consider to be core, but this might aswell be changed with something else that has a high bonus to MR. I just tend to like this over other items due to high MR and Health regen, for a relatively cheap price.


Usually my games don't get to this item, but against a standard team, i'd take this as my last item (cause it builds from Philosopher's Stone)


If you are against a high CC enemy team you might want to pick this over Shurelya's Battlesong, because of tenacity from this.
This is also build from Philosopher's Stone.

Cursed
Reducing Magic resist on Auto-attacks is nice, so remember to autoattack, whenever possible


Skill Shot. I usually save this ability, and tend to initiate with Flash instead of this. This gives me the option of using this to catching enemies later in the fight.
Remember that you are pulled to the first thing it touches, and thus can't be cast cross minions.
It can be used to cross walls in the jungle.


Toggle this, and try to stay near as many enemies as possible, this will do a lot of work for you.
Has a pretty high mana cost tho. If timed correctly you can use tics from this to clear creep-waves.


The active of this ability deals magic damage to nearby enemies, and the passive reduces damage to you (bread and butter of the little weeping mummy)


AoE stun. Generally don't use it if you can't catch 3 or more enemies in it.
Sometimes it can be wise to use this on fewer than three enemies, but generally isn't (due to the long cooldown). Be smart, and use it when it awards you the most.

1v1: Despair > Flash/Run > Tantrum (at this point enemy usually runs away, so i chase him, keeping up Despair and using Tantrum and bandagetoss whenever possible).

Teamfight: Hopefully you can wait with the initiation til the enemy team is piled, and then Despair > Flash > Curse of the Sad Mummy > Tantrum > Tantrum

This isn't a static route, and there's probably more effective routes, but I use this cause I find it the easiest.
The further you get in the game, the more you should mix it up, and the more often you should gank, but here is how I do things, the first time around.

Wraith Camp: Start off by smiting the big wraith, when the small ones get to you, pop a potion and Tantrum / Autoattack them to death.

Wolf Camp: Kill the big wolf first, and the small ones after, you should hit level two during this fight.
Pop a potion if you find yourself in trouble.

Doble Golem Camp: Autoattack one golem, pop a potion, (smite can be used when his health is around 500) Autoattack/ Tantrum second golem to death.
Level 3 should be achieved here.

Gank/Counter Jungling/Wraiths: If you used smite during the last fight, you need too find something to do until it gets up again one option is ganking, other option is stealing some enemy jugle - or if you've been a bit slow, do your own wraith camp again.
This will, if timed with teammates, net you a kill or assist.
If bottom isn't gankable, consider ganking mid, or even top.

Blue Buff: When you get here, you should be level 3-4 and have Tantrumand Despair as your skills.. Start out by hitting the big golem, popping a potion and activating despair.
After 3 Auto Attacks on the big golem, start using Tantrum.
When the small creatures die, de-activate Despair, but keep using AA's and Tantrum

Finishing the Jungle: After you killed the blue buff, you wanna clear the respawned parts of the jungle, all the way down to red buff, where you kill it (same approach as explained with blue buff).

- You should be level 6 by now, and this gives you three options.
Option 1, Ganking: Look around the map, if an opponent is stretched, or if a teammate is having trouble, throw a gank their way. This should net you a kill/assist, and a happy teammate.
Option 2, Dragon: If you have at least two spare potions, and a decreasing CD on Smite (around 15-20 seconds from the getgo), you should be able to do dragon, which gives you money and happy teammates.. If you are new to this, consider bringing along a teammate, who can help you and increase chances of succes.
Option 3, Buying ****: If you're low on health, or if none of the other options appeal to you, go back and buy your items (boots and philosopher's stone for example) and some potions, and restart your jungling route.

One of the most important things to note as a tank, and especially with amumu, is that your K/D ratio doesn't really matter. Your number of assists and your teammates K/D ratios is what really matter.

Your job lategame pretty much consists of setting your team up for kills. For example, popping on top of the entire enemy team with CotSM, might not get YOU a kill. But if your team isn't totally daft you could get everything from 1 kill to an ace, just by you crippling them with AEdmg/Stuns.

List:
Soraka
Fiddlesticks
Nunu & Willump
Katarina
Kennen

Explanation:
The math on most of these is Ultimate + Ultimate = Mayhem. Which is pretty self-explanatory imo.
Soraka was added to the list 'cause she does a very good job of keeping tanks alive.

Take into account that this is a guide for Newbies by Newbies. Feel free to comment and suggest.
Hope i explained myself without ranting too much. Try it out and have fun, might even work for you aswell.

Edits:
19/11/2011 - Updated for season 2, and transformed into a jungle/tank build
23/09/2011 - Added Nunu and Katarina to the "Buddy List", fixed minor "bugs"