It's called Ryokan or Minshuku for Japanese style accommodation in Japanese. Minshuku is more casual.
It's a bit different system from western hotels. Charge is not based on a room but a person and normally includes dinner and breakfast.
Some Ryokan or Minshuku don't like single occupancy as it's planned for two or more. The room is filled with tatami floor mats and you sleep on futon which is laid on tatami floor. Stuffs come and prepare futons while you eat dinner and clean up while you eat breakfast. Sometimes you have dinner in your room. You may have a private bathroom in your room but normally Ryokan / Minshuku has a public bath for all guests.
It's different from one you have been familiar with but this is the cultural experience and we recommend you to spend at least a few nights out of your stay. You can touch Japanese omotenashi.