This is the way I like to play Amumu. I know it's not perfect and this build is meant to be played against a balanced team. I'm writing this guide because I have had a ton of success playing Amumu this way, and I thought I would try to help others out and see what they think about it.This is my first guide, so I hope all goes well.

Marks- Magic Pen. Plain and simple, all of Amumu's abilities are magic damage and for a tank can be quite powerful.
Seals- Flat armor. I like to run armor because I'm a tank. Health per level runes would also work for Amumu.
Glyphs- Cooldown reduction. Amumu's abilities have quite long cooldowns so it's nice to have them be available whenever possible.
Quints- Flat HP. I find these to be the best for Amumu. Helps out a lot early game.

Exhaust-I love exhaust with Amumu because he is always gonna be in the middle of the battle. With exhaust, Amumu can easily make a dps carry useless in a team fight.

Ignite- I also love ignite. Helps get early game kills and it is especially nice if there is a good health regen champ on the other team, such as Dr. Mundo, Vladamir, or even a good Swain. Initiate with your Bandage Toss, Ult, and ignite the health regen champ or finish off a champ near death.

I like to start off with a Cloth Armor and 5x Health Pots. This combined with Amumu's E ability helps you stay in lane for a long time and farm quite nicely. Next, I like to go grab my Boots of Mobility when I have the chance. You can go with Boots of Swiftness as well, but I like the +5 to movement speed to help with getting back to lane and ganking. After that, I finish off my Aegis of the Legion. This helps out lane partners and teammates in team battles. Next, I grab my Sunfire Cape. With this, your W and E, farming becomes very easy and Amumu becomes a great AoE champ. Then, I like to pick up Banshee's Veil for the good amount of MR, life, mana, and spell block. Frozen Heart comes after, obviously building it throughout the game by buying Glacial Shroud earlier in the game. Frozen Heart helps Amumu so much by giving him plenty of mana, cooldown reduction, and lowers nearby enemy's attack speed by 20% to cripple their dps carry. Last, I like to grab Guardian Angel only if I'm being focused. If you are not dying, buy either Force of Nature, Thornmail, or Abyssal Scepter. If you really want to, you could substitute in a Rylai's Crystal Scepter to give your W and E a nice slow.

Pros:

Amazing tank.
Great initiator (possibly the best in the game).
Great farmer.
Can dish out a good amount of damage for a tank.
Highly underrated and overlooked.

Cons:

Can be a failure if you can't land his skill shot stun.
Ultimate needs to be timed very well to be effective.
People often complain about you stealing their kills, intentional or not.
Can cost the team the entire game if not played well.

Runes:
Marks- Magic Pen.
Seals- Armor or Health per level.
Glyphs- Cooldown reduction.
Quints- Flat HP

Summoner Skills:
Exhaust- Great disable against dps carry.
Ignite- Great finishing move, or health regen prevention.
Ghost- Good for chasing. I don't recommend.
Flash- Helps initiate. I only recommend if you have trouble hitting the max amount of enemies with your ultimate.
Fortify- Helpful to protect turrets early game. I don't recommend on Amumu because their are far more useful spells, and fortify loses a lot of effectiveness later in the game.

Game Phases:

Early game- Lots of farming. Lane fight if you are partnered with a champ who has good CC, such as Annie or Trynd. Maybe a gank.
Mid game- More farming, lots of ganks, team fighs, push lanes.
Late game- Less ganking, more team fights, Baron, push hard until inhibs and nexus are dead FTW!!!

Special thanks to Thee AC for being the best lane partner I have ever had. It's always hilarious owning with Amumu and Annie in a lane.

Thanks for your time. If you like my guide, please give it a thumbs up. If you had a problem with it, please leave comments on how you think the guide could be better. Even if you did like it, but saw some flaws, feel free to help me out. This is my first guide so I won't be surprised if I screwed up somewhere along the way.