The entire point of this build, which I originally called "Lean on Me" Amumu, is to ensure that enemies cannot outspeed your team, while also building huge amounts of durability. This build is meant to lane with a carry, preferably one specializing in magical damage output. Before you start laning, makes sure your team understands two things:
1: If the tank commits to attacking when the odds are hopeless, the rest of the team should retreat.
2: If the the tank commits to a dive, the team should dive with him/her.

Ghost- I take ghost for a few simple reasons:
1: It allows for much faster escapes in team fights once his part of the fight is over.
2: It allows him to get back into lane faster before he gets boots.
3: It allows him to escape VERY quickly after turret diving to help his carry (or carries) to get a kill.
Clarity- Amumu's Despair isn't very costly in team fights, nor is his tantrum. However, the way Amumu's passive allows him to feed Mage Carries very well makes it ideal to lane with one, and many of them use mana, will run out sometimes. Most mage carries have the sense to take clarity, but others either don't know what they're doing, spam their spells when harassing, or just have very mana costly spells. Taking Clarity gives Amumu's paired carry a back up stash for when they need it, as well as supplying him with some extra sustain.

For marks, as you can see, I get attack speed. I considered getting armor marks instead, but what it came down to is this- I can get lots of triggers on my passive if I take attack speed instead, which also allows me to get more from the runes, as attack speed is mark-centered. For seals, I build armor per level. This helps provide a decent amount of armor late game, and still gives a decent amount at the beginning for while I don't have any armor items. For glyphs, I build magic resistance per level. The reason for doing so is simply that all of the main items I should be taking add health, which makes me vulnerable to carries who take Maldred's Bloodrazor, and carries with percentage outputs already, such as Kog'Maw and Malzahar. However, usually percent damage is magic damage, and the time where they will be most effective is against Amumu in late game, so I build per level. The only downside to this build in this area is that Vayne will chop through my health anyway, because she deals true damage with her Silver Bolts, which do 8% plus a flat amount when maxed. Finally, for my Quintessences, I take Movement speed, which greatly increases my ability to chase in order to slow the enemies in time for my team to catch them. It also allows me a little bit for flexibility in lane while I still don't have boots, which will be discussed in the Items Chapter.

Expanded Mind- I chose to take expanded mind because it will A: allow me to use despair for longer periods of time, and B: give me more benefit than good hands. A good tank doesn't die much.
Summoner's Insight- Improved recall? It doesn't really do anything for Amumu. Who DOES it do anything for? Also, I AM taking two summoner spells that benefit from this mastery, so there is that as well.
Swiftness- Gives me insane movement speed when added in with Quintessences and my boots.
Meditation- Basically the same reason as I take Expanded Mind mana-wise. One might consider taking three points in greed instead, as Amumu will not have much gold income until the carry in your lane has been sufficiently fed.
Hardiness- Provides armor that my runes don't during the early parts of the game.
Tough Skin- Mostly because even with the sustain that is built in early, Amumu will still take some damage from the tons of minions that might swarm him when diving, due to the insane amounts of aggression that his abilities produce. Even with the passive of his tantrum, the damage reduction is crucial.
Vigor- I take vigor to work with Tough Skin and his Force of Nature near a games end in order to help escape if the enemy team runs lots of tank killers, ESPECIALLY Vayne.
Bladed Armor- This allows me to walk into a minion wave, use tantrum, and help get the minions low enough for the carry in my lane to last hit them. As a tank, that is the main job you will have in laning.
Indomitable- Same reason as tough skin, in addition to reducing flat damage output from swarms in a team fight.
Evasion, single point- Amumu will be getting plenty of Health from items before he even gets to building his boots. There isn't a point in taking Durability, much less Veteran's Scars. Plus, the point in it grants access to higher level masteries.
Initiator- Same reason for swiftness and Movement Quintessences. See above, or the Rune Chapter.
Enlightenment, single point- Allows access to high level masteries, and most of the time Amumu is not a champion with which to be singling turrets, so siege commander isn't necesary.
Mercenary- VERY IMPORTANT. Mercenary allows Amumu to get fed, even though he doesn't have gold quintessences in addition to his gold/5 items. These two things allow Amumu to become a fed tank, hence "carry as a tank." Tanks LIVE ON assists.

First Build: Regrowth Pendant, Health Potion.
The regrowth pendant is the ingredient to the philosopher's stone he should get first, as health is his primary concern. When he builds like this, it allows him to sustain next to his carry for long periods of time, and to hold towers unless dived by a tank-carry-duo.
Second buy: Philosopher's Stone.
The Philosopher's Stone is a gold/5 item, so this allows Amumu to get a decent amount of cash without taking gold from his carry. In addition, the philosopher's stone provides a sold amount of sustain, as well as being an item that builds into his final item, Shurelya's Reverie, which means that this gold/5 item will not be sold, which would put Amumu down by 400 gold.
Third Buy: The Heart of Gold
The Heart of Gold is another gold/5 item, which you will sell late game, but by the time you do, the gold/5 on it will easily have gotten you more than you lose from it. It also provides a solid amount of health which allows you to tank turrets longer when diving before you build your big health items.
Fourth Buy: Mercury's Treads
If the enemy team doesn't have more than one source of magic damage, take boots of swiftness instead. Movement speed is this play style's best friend. However, Mercury treads has tenacity, so when faced with lots of crowd control, it works out to roughly the same as swiftness boots in terms of speed, but also providing some durability against magic damage. Since you have lots of speed already at level one, don't worry if you get your boots later then most champions. You need the gold per 5 more than they do, and since you have good movement already, you have lots of flexibility in this area.
Fifth Buy: Rylia's Crystal Scepter (seems silly at first glance, but don't dismiss it)
Rylia's Crystal Scepter provides a solid amount of health, as it builds from the giant's belt, in addition to it's passive, which is what makes it your first most important buy. The slow, while only 15%, does get applied by Amumu's Despair, and Tantrum, AND Bandage toss (which does 35%, as it is a single target spell). This allows you to bandage toss them, get s 35% slow on them, and then 15% still after the bandage toss slow wears off. While the Bandage Toss is on Cooldown, you can still get them with tantrum if they only get slightly beyond range of your despair. In addition to all of this, Rylia's also gives 80 ability power, which means that he can successfully take down tanks with much more ease than before.
Sixth Buy: Frozen Mallet (seems less silly once you read Rylia's)
Frozen mallet gives even more health than the scepter, and gives an even larger slow than the bandage toss's rylia's debuff. The only ways to really escape once both slows have been bought are by using flash and flash-like abilities while the bandage toss is on cooldown.
Seventh Buy- Force of Nature
The Force of Nature is used in this build to provide speed, magic resistance, and health regeneration. This item will make you very difficult to kill by pretty much everyone except for Vayne (For this reason, go on the offensive when faced with her, because she is generally much squishier than you. However, in ranked gameplay, just ban her.). The magic resistance and % health regeneration will make you almost impossible to kill by % damagers.
8th Buy- Randuin's Omen
The Randuin's Omen builds from the Heart of gold, which saves 400 gold that would be lost if you were to sell it. It provides a passive that makes AD carries less of a problem, as well as providing great armor. The icing on this cake of an item is the active. When you initiate, hit r, w, and start spamming your e, you will love this item's active if you hit it right after your snare wears off.
Final Item- Shurelya's Reverie
Between Shurelya's Reverie's active, your massive speed, and ghost, there is almost no match to your speed. You are the bane of mobility champions. The Shurelya's Reverie also supplies some cooldown reduction, which makes you tantrum much more often, as well as supplying some sustain and health.

The build sequence for Amumu in this build is meant to provide high durability as soon as level 4, large area damage for good damage in team fights, and a quick initiation spell at first level.

Amumu being played as a tank shouldn't farm while he is in lane with a carry. When the carry dies (which they shouldn't when you are tanking) or recalls to base, that is the time to farm. Get to the place where your despair hits as many minions as you can, turn it on, and spam tantrums. The minions will target you due to the large aggression generated by the spells, and the attacks on you will reduce the cooldown of tantrum, allowing you to spam it faster. Once the carry returns, go back to preparing minions for your carry to last hit. The gold/5 items and mercenary (and the 3 points in greed that one might take) will allow you to still have sufficient gold for building items. As a tank, don't worry too much if you end up with a large number of deaths but not many kills. Getting fed from kills is the job of the carries- be more concerned about getting large numbers of assists, as they will dramatically increase your income thanks to your points in mercenary.

In a team fight, be sure to focus the other tanks. Your percentage damage from despair and magic resistance reduction from your passive will allow you to reduce their health much faster than many tanks will be able to hurt you. When the enemy team starts to get near half-health, drop your curse of the sad mummy so they cannot run. This should be a signal for the rest of the team to drop all their big damaging spells on the enemy, and probably win the team fight.

Some of the statistics in the above information are incorrect, and I have taken the liberty of calculating them for you. You're welcome =)

Attack Speed:
roughly 1.071/second

Movement:
Health above 70%, no shurelya's buff- roughly 474
Health above 70%, with buff- roughly 664
Health below 70%, with buff- roughly 460
Health below 70%, no buff- roughly 445
Health above 70%, no shurelya's buff, with ghost- roughly 640
Health above 70%, with buff, with ghost- roughly 896
Health below 70%, with buff, with ghost- roughly 621
Health below 70%, no buff, with ghost- roughly 601