If it has no back and no arms, it is a stool or a bench and you sit on it. If it has a back and arms, it is a chair or an armchair and you sit in it. If it has a back and no arms, it is a chair, but what do you say? I would say "in". I cannot think of any difference between "in" and "on" due to: size or design of chair; situation (relaxing in living room, eating in dining room, learning in a classroom); or audience (child or adult). I have heard some people say "on". It may be a small regional difference in how people talk.

I see William Fay's distinction. You sit in a chair and you enjoy the act of sitting for a while. That is where you are. If someone (you or someone else) is concerned about you placing yourself on the seat of the chair, then it is "on", on the seat of the chair like on the stool. In a museum of old furniture, the sign says, "Do not sit on the chair." When the fabric is delicate and ornate and you are concerned about damaging it, you ask your friend, "May I sit on this chair?"

Both are correct. When you speak about sitting on a chair, it’s more about that the chair may have very delicate fabric or you might be worried about getting it dirty and you’re asking permission. Sitting in a chair implies more that you were going to be in a place for a while.