Amumu
Don’t Throw a Tantrum
Lane harassment and consistent damage output.
Resolve
Durability and crowd control
AftershockAfter
immobilizing an enemy champion gain defenses and later deal a burst of magic damage around you.
Font of LifeImpairing the movement of an enemy champion heals nearby allied champions.
ConditioningAfter 12 min gain +8 Armor and +8 Magic Resist and increase your Armor and Magic Resist by 3%.
OvergrowthGain permanent max health when minions or monsters die near you.
Sorcery
Empowered abilities and resource manipulation
TranscendenceGain bonuses upon reaching the following levels:
Level 5: +5Ability Haste
Level 8: +5Ability Haste
Level 11: On Champion takedown, reduce the remaining cooldown of basic abilities by 20%.
WaterwalkingGain
MS and AP or AD,adaptive in the river.
Adaptive Force
+9
Adaptive Force 
Health Scaling
+10-180 Health (based on level)
Why Aftershock
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.
Why Font of Life
As Amumu your [R] has the opportunity to provide a ton of healing in a teamfight if you're able to hit multiple targets.
Why Conditioning
Conditioning on Amumu allows you to scale into the mid game as a massive tank alongside Aftershock and Overgrowth.
Why 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.
WhyTranscendence
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.
WhyWaterwalking
Waterwalking empowers your ability to duel in the river for neutral objectives like Dragon and scuttle crabs.