Tank engage and teamfight utility.
Durability and crowd control
After immobilizing an enemy champion gain defenses and later deal a burst of adaptive damage around you.
Impairing the movement of an enemy champion marks them. Your allies heal when attacking champion's you've marked.
After 10 minutes gain 8 Armor and 8 Magic Resist and increase your total Armor and Magic Resist by 5%.
Gain permanent health when enemy minions or monsters die near you.
Empowered abilities and resource manipulation
Gain 10% CDR when you reach level 10. Excess CDR becomes AP or AD, Adaptive.
Gain MS and AP or AD, Adaptive when in the river.
+9% Attack Speed
+5 Armor or 6 MR (match-up)
Aftershock is a great rune for entering a gank with [Q] providing you the resists to survive and a bonus burst of damage with the explosion.
As Amumu your [R] has the opportunity to provide a ton of healing in a teamfight if you're able to hit multiple targets.
Conditioning on Amumu allows you to scale into the mid game as a massive tank alongside Aftershock and Overgrowth.
Overgrowth is likely the best rune in this slot primary due to Amumu's fast clear speeds and lack of major value from the other 2 options.
Transcendence is a great rune choice on Amumu as he likes to build all of the CDR tanks items, allowing him to go over cap and still make great use of the extra AP gained.
Waterwalking empowers your ability to duel in the river for neutral objectives like Dragon and scuttle crabs.