FFX: Perseverance Achievement
Penance is the most difficult fight in the game when done legit (meaning you fully play out the battle). You will want to have completed "Perfect Sphere Master" first, hopefully in the most efficient way to get your characters as close to their max stats as possible. It also helps to have a very high luck stat (120+) which is normally a stat most people will pass on.
To unlock this battle, you must first clear all of the Dark Aeons (for more info on these, check here). From there, you can travel to the fight via the Airship location options. Check here for the best setup and method of tackling this boss.
The other method to defeating him utilizes the Yojimbo summon. As mentioned in the Road Map, using this summon properly to get an auto-kill effect can be somewhat confusing, though it will obviously knock the boss out immediately. Either way, once the boss is defeated the achievement will unlock.