Hi, iTryToBeCool. Amumu was one of the first champions I played, because when I started LoL I knew that i'd be more into tanks, because I like having high survivability. The build that I use has changed a lot over the months that I've been playing, and a goal of mines has always been to make an Amumu guide to return the favor, since I've literally looked at every single recommended/high rated Amumu guide and tried them.

Build one: This is what I used to play as, and I think it works well. It uses the items most commonly built on Amumu, with the exception of a death cap. High HP, and High AP. You're not a good tank if you just get ignored because you can't do damage. You need to be able to go in there and make people fear you!

Build two: Tanky Jungle Amumu is tanky and a very strong ganker.

Don't down vote without trying it. Please.

Build 1: Spell Pen Marks to do da damages. I wouldn't suggest changing this. Seals for defense, survivablity, tankyness, All that. Again. I wouldn't suggest changing that, either. Glyphs of cooldown reduction. I like spamming my skills after grabbing a blue buff or something. AP Runes might be better, but when I wrote this I didn't have them.

Build 2: Again, Spell pen to do da damages. I tries using attack speed marks, and it isn't oobad. It helps improve your jungle time more than marks do early on. Seals of defense. They're cool in the jungle, and finally, Ap Glyphs. That AP actually goes a long way. It's 25 more damage on your bandage toss and ult. This isn't AP Amumu, so you need some help in that department.

Build one: Amumu is a tank. He needs his 22 points in defense mastery, or he might actually become a squishy unviable AP carry. You might be wondering why I don't take the 10% spellpen. It's because you won't be needing it. You have 28.5 with Sorc shoes, and Abyssal Scepter removes 20 MR from them, which is the same thing as 20 spellpen. Percentage-based spellpen is applied after normal spellpen, meaning, that if they have 100 MR, it's only 5 more damage.

Build 2: Typical jungler defense masteries, with the other nine points being dumped in utility for the movement speed and buff duration.

Build 1: Anything after Rabdon's Death Cap is completely okay, as long as it's Tanky or AP. The shoes you buy should be anything except Berserker's Greaves.

Build 2: You don't need Ninja Tabi. I only take them because I already have a cloth armor and it's the cheapest thing available. I don't want to be running around the map like a sack of ****. I find Amumu to be really slow, and that's why I take movement speed quints and masteries. I would suggest following this purchase guide as well as you can, and afterwards, you will have 2 free slot. Some of the better items to build are Warmogs, Rylai's, Abyssal Scepter, and other tanky/ap items. If the enemy team is AD heavy, thornmail is cool, and if they're AP heavy, A banshee's is awesome.

Build 1/Laner:
In the current meta, Amumu has no place in lane except for in the jungle. This doesn't mean that you still can't pick him and do decent. Amumu only needs to be in his lane for a few minutes, until he's farmed up and ready to roam. For the first few minutes, just play defensively and farm. You can use your skills to farm. It doesn't matter, because you're not going to be trying to fight your opponents. You're not going to get destroyed by them, but you won't win until your damage output get higher. But don't not harass. You can bandage toss to them with tears on, use your tantrum, and walk away like if nothing happened. Once you're farmed, go gank. It's your job to CC the enemy and let your team mates pick up kills. Sometimes you'll get the kill. No problem. A fed Amumu is a strong Amumu. Once your AP is high enough, your bandage toss and ult together should be taking away half of your targets health. I'm not even exaggerating. If you chain your skills with tears on, they will not be able to escape with much health remaining.

Jungler:
It's not rocket science. Start at blue, do dogs, Wraiths, red, golems, wraiths, dogs, golems, and just keep on killing them as they spawn. Don't worry about ganking. If you have to gank urgently, you can put a point in Q before level 5. Ammumu basically clears the jungle. Again and again. You should go back when you have enough gold to get your philosopher's stone. If you don't have 800 gold, don't go back. I usually go back with ~1200 for the stone with boots and a ward for my team. It's the same thing, though. Gank as hard as you can. Don't be afraid to turret dive if they're low and you're high. Just make sure your team will commit.

Pretty skimpy guide, but I don't really know what to say. Remember to buy wards, don't be scared to tank a tower here and there, enjoy your stupidly high HP regen on build 2, and check out this guide. I followed it when I was a little boy only learning how to play this very fun and intricate game. http://www.mobafire.com/league-of-legends/build/amumu-the-tank-slayer-46190