Passive: Cursed Touch
It's a pretty helpful passive if you remember to utilize it fully, make sure to always melee enemies when possible to burn through some of the MR they have.
Q: Bandage Toss
This skill is what I feel makes Amumu a decent jungler. Obviously his AoE skillset gives him a decent clear but without this skill, his ganking potential would be so bad. The long range of it can really catch enemies off guard, and I can't remember the amount of times I've been able to bandage to a creep in the jungle to escape a nasty gank.
I take this at level 4 usually unless an enemy is just asking to be ganked at level 3. I max it last as its main use is as a gap closer and it offers no benefits within the jungle.
As an ability which does AoE percent damage, it can WRECK a lot of enemies that stack health. Also if you have a Rylai's and have this active, the enemies aren't going anywhere fast. I take this at level 1 rather than tantrum because from my tests, it gives a faster start.
With its cooldown on being hit effect, this ability is up a lot of the time while you're in the jungle. I max this early on because when you gank, the enemy isn't going to stick around long enough for Despair to do as much damage.
R: Curse of the Sad Mummy
This ability is just Amumu. This ability is what gives him a HUGE presence in lanes as soon as he hits level 6. If your laner doesn't completly derp, you can walk into lane, use this, and walk away with an easy kill in hand.
Late game this skill just gets even better! Enemy all in baron or dragon pit? everyone having a mexican stand off? Enemy team thinking it's a good idea to walk into that bush? If you answered yes to any of these questions press 'R'!
Obviously this ability is a little bit more than that but you can see the point, the only thing you have to make sure you do is focus the damn carries. I've been in so many games where someone else has instalocked Amumu then wastes there ultimate on a support and tank while the enemy carries just beat them to a pulp so don't be that guy!