The characters in the game can be broken into 3 categories: slaves, assistants, and NPCs. Slaves are who you train throughout the game. Assistants are helpers that give you advice and support. NPCs are people that you encounter throughout the game.
Slaves are girls, from various animes and games, who you train through the game. Once you have created your character, you will simply continue to train slaves for the rest of the game. Each slave has unique starting stats, events, endings, etc. Thus, you can get vastly different playing experiences with different slaves.
Assistants are people that help you train your slave. Although they show up during some events, you cannot interact with them (i.e., ask them questions).
You can change your assistant from the Options menu. The default assistant is Shampoo and she will become your assistant if you try to set an invalid assistant. Some slaves, like Belldandy and Urd, have preset assistants. Genma is special in that he only appears when you train Shampoo and have her set as your assistant. Some slaves that you've trained can also be used as assistant.
NPCs are characters that you will encounter throughout the game. Some, like shop keepers, have static locations and will sell you Items or pay you for doing Jobs. However, they're important because you can randomly encounter them during events, such as when you Take a Walk, and gain rare Items and stat changes.
See the NPCs category for more information on a specific NPC, because there are far too many to list.