Eliminate the Act III final boss - Azmodan - while accomplishing certain tasks during the fight. They can be done over as many playthroughs as you want.
No Reinforcements - Destroy the Demon Gates before any Demon Troopers can spawn while fighting Azmodan on Hell difficulty or higher
The demon gates are the pillar-type objects that appear during the fight, and you must quickly destroy them before any enemies spawn from them.
Hot Foot - Kill Azmodan without walking over the bloody demonic ground on Hell difficulty or higher
The bloody demonic ground will slowly grow from the outside ring of the arena you fight Azmodan in (primarily, directly behind where he starts out), so either beat Azmodan before it becomes a problem, or keep moving Azmodan toward the middle of the arena.
Be Cool - Kill Azmodan without getting hit by a fireball on Hell difficulty or higher
Azmodan will occasionally launch huge, slowing moving (but heat-seeking) fireballs at you that you must run from to avoid. These are generally easy to dodge as a ranged class, given the increased distance between you and where they appear, giving you a bit more time to react.
Azmodan Adds - Kill Azmodan without killing any of his minions on Hell difficulty or higher
This will need to either be done alongside No Reinforcements, or if you fail, you must simply ignore the add(s) for the duration of the fight and damage only Azmodan.