Amumu is a damage dealing tank that locks down high priority damage dealers. The amount of AOE you deal is amazing, so you wanna throw yourself into the fight as hard as possible.
Pros

High damage output in late game
Takes a lot of hit
Fast camp clear post six

Cons

Can be counterjungled easily
Weak counterjungle
Ganke potential is not amazing

Check the notes to see the explanation of items, ability order, runes, or masteries.

Start gromp, go get your blue and red, and then back if you have enough health, or move to krugs or scuttle. I do not recommend ganking, because you are extremely weak early game. After you back, go to gromp or krugs and just rotate through the jungle, killing wolves and raptors as well this time. If you see a lane pushed, go to gank AFTER you hit six.

Some people want to walk in and immediately throw their bandage toss. I am here to say that is a viable way of ganking. However, it is better is you walk into lane and try to zone your opponent to a place you could easily land a bandage toss. If you get closs enough ult, and allow your laner to catch up. If they flash, either flash and q or just q. Be sure you know how far you can chase, because you don't want to give up a kill early in the game.

Unless the enemy jungler is stupid I do not recommend counter jungling. If you are counter jungled, ping for assistance and run away, unless you are a level up and near the same amount of health as them. Unless, of course, you are against some assassin or high damage dealing jungler, like Nidalee. Flash if you have to, and look for a creep to bandage toss to.

You want to initiate teamfights as much as possible. Simply bandage toss in, turn on your w, walk past the enemy tank to between the backline and tanks and ult. This allows your damage dealers to kill the tanks or make them flee, and quickly move to the damage dealers. By now, you should have your bandage toss. Save it to lock down fleeing enemies or ones that have been zoned away from their allies. If it goes badly, you may need to bandage toss chasing enemies and flash away, or just sacrafice yourself for the greater good. Bear in mind that this is all situational.