How to get the Asura Summon in FF4
The first optional summon that you can get in Final Fantasy IV is Asura. She is located in The Land of Summons which is reached through The Cave of Summons, a dungeon that's found in the Underworld. For more information about how to reach this location I recommend you check out my FF4 Walkthrough Chapter 9. It'll show you maps and explain how to get here.
As for the fight with Asura herself, depending on your character's levels, this will be one of the toughest fights yet. Rosa should be at least level 36 so she learns the Reflect spell. This spell is required for the easiest strategy to defeat this boss. When I did this boss, I was almost level 50 on my entire team.
If you don't have reflect, or if you'd just like to use a different strategy, you can Silence Asura by using weapons that proc the effect. Asura is immune to Silence except when it procs off of an item such as Mute Arrows, Magemasher or Rune Staff.
For the most part Asura will not attack you unless you attack her first. Use this to your advantage if your team is under leveled, you can spend many of the first rounds of the fight casting beneficial spells like Protect and Blink. Since our strategy involves Reflect, you'll want to avoid using any of Rydia's Black Magic. I would recommend you just Defend on her each round, that'll keep her alive incase she gets hit too.
Here is a quick list of all Asura's heads and what each of them does. So long as you cast Reflect on her, you won't have to worry about much of the spells mentioned below.
Blue Head: Life
Red Head: Protect
Flesh Head: Curaja/Cure
Once Asura is defeated you'll obtain her as a summon and you'll also be able to challenge Leviathan. He's the old man that was standing next to Asura in the same room. Speak with him when you're ready for that battle. (Return to Land of Summons first and use a Cottage! Then challenge Leviathan)