I first thought you meant music (a dedicated contributor found all of the original themes a few posts back), but as you obviously mean story, the Grim Reaper will tell you all of their backgrounds but he'll be pretty subjective about it - each Domina also has her own views on the next one. I take it you also meant how to fight them and unfortunately, I'm not much of a fighter, but many have shared their own tactics all over the thread.
Or did you just mean how they fight and what they fight with?
Chicken:
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. (No, seriously, Dekarous did leave a Chicken placeholder.)
Katrina: Attacks with Physical. Resists weak poison.
Madeline: Attacks with Wind. Resists Poison. Fire does 1.5x damage to her.
Narcissa: Attacks with Poison. Immune to all Poison-based states. Ice does 1.5x damage to her, 1.2x when in Trance form.
Mary: Attacks with Darkness. Resists Poison. Light does 3x damage to her. Normally requires Llavanya's help to be slain.
Rosita: Attacks with Earth. Darkness and Thunder both deal 1.5x damage to her. Resists weak poison.
Franziska: Non-elemental attack. Darkness deals 1.5x damage to her.
Meke (Mika and Maki): Non-elemental attacks. Attacks twice. Fire and Thunder deal 1.25x and 1.5x damage to "them" respectively.