You can see Moetans by selecting the Collection button on the right side of the main screen.

Each Moetan has a certain bonus, which increases in power the more you level it up. Bonuses include extra gold drops from Monsters, extra Damage power, extra Moe Crystals when using Reborn, etc.

To see what bonus each Moetan has, you can check the Moetan menu. (see menu FAQ)

Affection or climax points are used as a currency used when tapping on your Moetans in order to increase their intimacy level.