Ereshkigal is the final boss of the Ultimate Lair in Lightning Returns: Final Fantasy XIII. Beating him will earn you two gold trophies:
Monster Bane - Proof of mastering all creation and conquering even the Ultimate Lair. [Gold / 90G]
Holy Garb of Destruction - Proof of obtaining the ultimate garb, imbued with destructive magic. [Gold / 90G]

This boss is completely optional and missable! To get to the Ultimate Lair you have to beat all main story quests and complete a certain amount of side quests. Basically, you need 100 side quests points: 1-star missions are worth 2 points, 2-star missions are worth 4 points, 3-star missions are worth 8 points. So if you do around 30-40 quests of different difficulties, you should be good to go. The Ultimate Lair will then unlock on Day 13. Fight through all of its 33 levels and defeat as many Last Ones as possible along the way. They drop extremely strong equipment which will come in handy for this fight.

Now as far as the boss fight goes: Poison & Overclock are the key for a quick victory.
1. Use Magic attacks until Ereshkigal is staggered
2. Cast Deshell, Slow, Imperil, Poison
3. Keep spamming overclock, use Elixir or Ether when you run out of EP.

If you are going for all trophies / achievements, then you would probably have the Elixir at this point of the game (earned by selling 100 soul seeds to merchants). It will completely refill all of your EP, which guarantees a quick victory. There's also an Ether on floor 32 of the Ultimate Lair. So if you don't have an Elixir, at least get the Ether (restores 3 EP) and you'll be fine.

This video was captured on the first playthrough! However, this boss fight is significantly easier in New Game+.