The lights are on
The combat is turn based. Each demon has elements it's weak to, hitting a demon's weakness grants the attacker an extra turn, regardless of which side their on. So you have to keep in mind what the demons you put on your team are weak to and what types of skills they have. The key to beating most bosses is finding the right set of skills to use on them.
You recruit demons by convincing them to join you, or fusing the demons you already have. The game isn't designed for you to stick with the same team, you'll have to fuse off or delete your demons to get more useful ones. Fused demons can inherit skills from the demons used to make it, and some recent entries have let you pick which skills get inherited (such as IV, thankfully).
Outside of battles, you explore dungeons and make choices that decide your ending. You can buy weapons, armor and items.
I'm not good at explaining things, but I think I got the basics down.