First off, this build works best in the Twisted Treeline, or when Jungling. Since this build limits Amumu's role as a tank, and maximizes his ability to deal lots of damage (depending on how much maximum health the enemy has), he cannot jump into crowds of enemies or survive very against more than 2 enemy champions. This guide focuses on building on Cursed Touch.
This works very well against all sorts of tanks, yet still holds up strong against other champion types.
This build is effective on most champion types (especially tanks), offering fast kills, as well as very efficient Jungling.

08/01/12 PATCH AMUMU CHANGES:

Amumu

Cursed Touch (Passive)
Duration reduced to 3 seconds from 4
Tantrum
Mana cost reduced to 35 from 50
Bandage Toss
Ability Power ratio reduced to 0.7 from 1.0
Curse of the Sad Mummy
Ability Power ratio reduced to 0.8 from 1.0

Although I edited a few things in my guide, this does not make this build any less effective. Just expect a little less damage. Also, this will allow you to idly cast tantrum to fill Tear of the Goddess with less mana penalty.

Marks

For Marks, take 9 Greater Mark of Magic Penetration's. This is to take care of any remaining magic resist the enemy champion might have, and helps a lot in the early game. I don't see any reason for any substitutions, so leaving it at that is fine.

Glyphs

For Glyphs, take 9 Greater Glyph of Scaling Ability Power or 9 Greater Glyph of Scaling Cooldown Reduction's (or even a hybrid between the two).
If you like casting abilities a lot, you could easily substitute Greater Glyph of Scaling Cooldown Reduction. You won't have to worry about mana, as that is covered by Archangel's Staff. Any of the two will do, but personally, I prefer the Greater Glyph of Scaling Ability Power since both Bandage Toss and Curse of the Sad Mummy add 100% of your ability power to its extra damage.

Seals

Seals, the hardest choice for Runes, in my opinion. Since the Primary Seals focus on defensive stats, use this opportunity to take 9 Greater Seal of Resistance. Since Amumu is generally a tank, this will help him live up to that, and provide decent protection while not sacrificing any needed stats (for this build). Using these, you are safe against (in most circumstances) any PA DPS champion, such as Xin Zhao, Wukong, etc. I don't see any reason to substitute these out, as the minimal ability power on Greater Seal of Ability Power and Greater Seal of Scaling Ability Power won't be as useful as Greater Seal of Resistance.

Quintessences

Although it might seem odd when you initially see my rune setup, having 1 Greater Quintessence of Ability Power is very important to this build. The Greater Glyph of Scaling Ability Power's that will give you Ability power, won't help you as much early game (they don't even out with Greater Glyph of Ability Power until approximately level 8). You need to be able to hold your own and fight champions. Since this build focuses on Amumu's ability to lower magic resist, you need to be able to do something while that magic resist is lowered, or it will be useless. Having this (along with armor from Greater Seal of Resistance) enables you to take down enemy champions before they take you down (or, faster kills).
Take the Two Greater Quintessence of Scaling Ability Power for the same reasons as the Greater Glyph of Scaling Ability Power.

Summary

Take Greater Mark of Magic Penetration to deal with any remaining Magic resist, Greater Glyph of Scaling Ability Power for more damage output or Greater Seal of Scaling Cooldown Reduction if you prefer, Greater Seal of Resistance to be able to kill enemies before they kill you, and finally a single Greater Quintessence of Scaling Ability Power with two Greater Quintessence of Ability Power to further your Damage output even more, and still be able to be effective early game, as well as mid game.

Looking at my Masteries, your initial reaction will probably be asking why there is no point in Executioner
This is mainly because he doesn't really need it. You could have it, if you wanted to, but the point would better be used on something such as Meditation . Executioner only helps in certain situations on Mages. The main one being Veigar, with his 1000+ damage spells. With Amumu, his abilities won't get to the point where it will make a difference.
Now, other than that, this mastery book is setup to maximize Amumu's damage output using ability power, as well as provide him with some defense on Hardiness and Resistance since there is no better use for those points. The reason for having points on Expanded Mind is to further increase early mana, allowing Amumu's Despair to stay active longer, as well as increase the duration he can lane before needing to recall.
I slapped a point on Summoner's Insight to decrease Flash's CD, which I find useful. You could easily put that point on Alacrity , Durability , or Vigor if you either don't plan on using Flash as a Summoner spell, or don't feel like you need the 15 sec CDR.

If you want to, take Clarity as early game. It will help out with mana, but only in early game. I rarely find myself in situations where it is useful in late games, as Archangel's Staff provides any needed mana and mana regeneration.
I take heal first, and use Flash, mastering flash.

As for a second Spell, (if you plan on using clarity) anything form heal to exhaust) to Exhaust works just fine. Amumu doesn't need to use Clarity late game, with very few exceptions, and doesn't have any specific need for any of the other Summoner's spells. Go with what you feel you will need. Pick up Heal if you feel like you are going to need some backup, or help you with the strategy mentioned before with Clarity. I have flash listed, because Summoner's Insight reduces its CD by 15 seconds, so you don't need to waste a precious mastery point to get the most out of your secondary ability.

If you want to ensure you get the kill, or just no one else on your team has it, take Exhaust. Again, anything you feel necessary, take.
The only thing I don't recommend is getting Cleanse. Cleanse is a very useful skill, although rare for a champion to have. Amumu's Bandage Toss allows him to escape from tight situations when fleeing, leaving with that spell the moment an opponent uses Exhaust.

Order

(for quick reference)

Doran's Ring> Boots> Sorcerer's Shoes> Meki Pendant> Tear of the Goddess>
Amplifying Tome> Malady> Negatron Cloak> Blasting Wand> Archangel's Staff> Blasting Wand> Abyssal Mask> Needlessly Large Rod> Rabadon's Deathcap>Sell Doran's Ring> Frozen Heart

Now we come to items. The build, the aspect that defines the role of a Champion.
First off, Amumu needs mana regen to keep his Despair active. Therefore, starting with Doran's Ring will let you, again, stay in your lane longer, and minimizing how often you must Recall. The health is also very useful early game, and that 15 ability power is perfect for extra damage on Bandage Toss and Tantrum. Since this is a Twisted Treeline build, boots and three potions are not as necessary.
Second, buy Meki Pendant on after your second Recall (first being buying Sorcerer's Shoes). For the obvious reason, allowing you to stay until you have enough for Tear of the Goddess. Buying Tear of the Goddess immediately after buying your boots is very important. Since it gives bonus mana from casting spells, the earlier you buy it, the better.

Malady

After buying Malady is where this build really starts to come into its effectiveness. By this time, you'll probably have Despair at least level 3. This item, along with Cursed Touch, is very effective at maximizing the damage Despair does, generating increased damage output. This can bring an enemy's magic resistance below 0.
Malady's second passive adds 20 magic damage as an on-hit effect. This will allow you to punish your enemy even more than just your abilities with basic attacks, and since the extra damage is applied as magic damage, you won't have to worry about the majority of it being reduced.
Use your bandage toss to get close to apply the hits if you're having trouble using this item effectively.

Archangel's Staff

This item is very important, but don't buy it before getting a Negatron Cloak, unless you're laning against someone such as Riven who uses purely Physical Attacks.
The Mana Regen on this item is increased by 18 per 5 seconds compared to Tear of the Goddesss, which, along with the extra Ability Power, is very important to have as early as possible.

If you are having trouble staying alive at this point, definitely buy Thornmail. It will allow you to absorb more melee hits (obviously, only get this if you are having trouble against a melee user).

Abyssal Mask

The main reason for incorporating an Abyssal Scepter into this build is to greater reduce the enemy's Magic Resistance. This works well whether you are attacking enemies or not, allowing both you and your teammates to do increased damage with their magical damaging abilities. The Magic Resistance on this, as well as the ability power, is high as well, and adds more effect overall. This will also allow you to survive when dueling other casters. This, along with Malady, is one of the key items of this build, and there is no room for substitutions.

Rabadon's Deathcap

Next, we come to Rabadon's Deathcap, the primary element in any Mage. Since Amumu's Despair gives a hefty damage boost for ability power, Amumu needs this item. Without it, he would have an unusually low damage output with his abilities, and nothing to utilize his opponent's decreased Magic Resistance on. Since both Amumu's Bandage Toss and Curse of the Sad Mummy, as mentioned before, have high ability power ratios (Bandage toss at .7, his ult at .8), Amumu will be able to maximize the damage he can get with his abilities.

With the enemy's Magic Resistance being reduced, you can expect Amumu's abilities to do either full, or close to full damage, as their remaining Magic Resistance will barely combat to the high damage of both Despair and Curse of the Sad Mummy.
Note - don't expect bandage toss to do full damage if you plan to initiate with it.

Frozen Heart

Frozen Heart - First of all, do not sell Doran's Ring until you have enough money to at least buy Glacial Shroud. Also, if you are having a hard time surviving in early game or mid game, take this before purchasing Archangel's Staff or Rabadon's Deathcap respectively. The cooldown reduction this item gives will also dramatically increase your damage output, and can turn a seemingly hopeless situation into a great outcome.

There are several reasons that Frozen Heart is in this build. One of these is, of course, the hefty +99 armor bonus it gives. In most cases, the item you buy after selling the starting Doran's item usually is just to further the Champion's effectiveness and power. In this case, however, there is a different reason.
Frozen Heart is a key item for late game. Buy this time, the enemy will probably have gotten used to your build, strategies, and knows how to deal with you effectively. Frozen Heart adds a new dimension to the small, almost comfort zone your opponent has formed when facing you.

The Aura - This is a very important aspect of Frozen Heart. Reducing the opponent's attack speed, as well as providing armor, will significantly decrease the damage you take in a fight. You will still be doing the same damage, while your opponent has a significantly decreased, while also sudden, drop in their effectiveness.
Take advantage of your opponent's bliss after purchasing this item, and you will pick up some neat kills. It won't be something they were expecting.

The Cooldown Reduction - If you want Cooldown Reduction, (mainly for your ultimate, as well as Tantrum) earlier in the game, definitely buy Glacial Shroud before Negatron Cloak. The 15% cooldown Reduction on Glacial Shroud is very useful. In some situations, completing Frozen Heart's recipe earlier is the best thing to do.

Lastly, the mana. The reason other important reason for this item is to increase the extra ability power you receive from Archangel's Staff's 3% of your mana added to your ability power. For clear reasons, this is important.

Use Your Discretion!

Use your discretion in this build to allow it to have most effectiveness. If your opponent seems to be harassing you with Melee attacks, leading you to constantly have to recall, consider buying Frozen Heart. If you are having trouble farming minions or just need some extra damage with your abilities, buy Needlessly Large Rod earlier than you normally would.

Summary

Archangel's Staff for Mana, Mana Regen, and increased ability power depending on your maximum mana.
Malady to lower enemy's magic resistance along with Cursed Touch, and Abyssal Mask for the same reason. Also, Malady's extra 20 Magic Damage on basic attacks increases damage output dramatically.
Abyssal Mask and Rabadon's Deathcap provide the majority of your ability power, along with Archangel's Staff.
Frozen Heart makes a great impact, and can come in to play anytime during your game, and is used for its armor and CDR.

Overall

The concept of this build is to try to always have Despair active whenever enemies are near. (except, of course, when near a turret while an enemy champion is present). It focuses on allowing Amumu's damage output to be high, as well as any other mages who may be on your team.

I also understand that this build can work very well with Fiddlesticks.

Using Malady

Remember, Malady's effect triggers when you attack the enemy with your basic attacks, so don't rely completely on your abilities to do all the damage. After initially casting Tantrum, make sure your opponent has 4 stacks of Malady on them. Also, don't forget Malady's on hit passive effect, adding 20 bonus magic damage to your basic attacks on hit.
Basic attacks are just as important as casting your spells, so make sure to keep it constant.

This build works very well when Jungling. Just activate Despair, start off with Tantrum, and they'll die within a matter of seconds. I don't reccomend you start off jungling, as you are going to need several points in Despair, as well as some ability power.

Smite is not necessary, unless picking up the Golem Buff early-game is important to you.
Jungling on Amumu is, suprisingly, very efficient. He can take down any Epic monster in a matter of seconds, and can easily solo the Dragon in the Twisted Treeline, especially after obtaining Frozen Heart.
Tantrum also helps a great deal, due to its passive effect.

There is no need to read this if you have read through my whole guide, this is here for anyone who just wishes to get the main idea of my build. If the latter is the case, also make sure to read the strategy section if you plan to try it out.

Runes

Take Greater Mark of Magic Penetration to deal with any remaining Magic resist, Greater Glyph of Scaling Ability Power for more damage output or Greater Seal of Scaling Cooldown Reduction if you prefer, Greater Seal of Resistance to be able to kill enemies before they kill you, and finally a single Greater Quintessence of Scaling Ability Power with two Greater Quintessence of Ability Power to further your Damage output even more, and still be able to be effective early game, as well as mid game.

Masteries

20/6/4

Basically, look at the mastery page at the top. Give anything in the offensive tree that will add ability power, but don't take Executioner unless you need it. Give a point on Summoner's Insight to maximize Clarity's effectiveness.

Summoner Spells

[[Clarity]/ to increase the amount of time you can stay before Recalling
Secondary Spell - anything you prefer, the best being Flash, Exhaust, and Heal.

Items

Archangel's Staff for Mana, Mana Regen, and increased ability power depending on your maximum mana.
Malady to lower enemy's magic resistance along with Cursed Touch, and Abyssal Mask for the same reason. Malady also for its second passive (20 bonus Md on basic attacks)
Abyssal Mask and Rabadon's Deathcap provide the majority of your ability power, along with Archangel's Staff.
Frozen Heart makes a great impact, and can come in to play anytime during your game, and is used for its armor and CDR.