I remember from my English classes that you sit on a chair, but in an armchair (of course, unless you sit on the arms). It may be a regional thing, I don't know -I was taught British English, not American.
Anyway, Google gives twice as much results for "sit in a chair" than "sit on a chair".
I'm going to say you can use either. Maybe, instinctively, native speakers know which one to use depending on the type of chair involved.
A mother takes a child to the dentist. "Sit on that chair and wait, " she tells the child.
I'm being nagged to go out for a walk: "I don't want to, I'm so comfortable in this chair by the fire".
Now that I think about it, I think the type of chair IS significant. If I'm sitting by the fire, it's likely in an armchair, and I would never say "on an armchair". If I'm waiting in the dentist's surgery, I would never say I was sitting "in a chair".
^Indeed, in Spanish we use the preposition "en" regardless of what kind of object we are sitting on/in.
As far as I know, both "in" and "on" are correct in English, as some of you already said, and I agree it depends greatly on the context. It definitely depends on what kind of seat we're talking about, but my guess is there is also a nuance there about how comfortable we are - personally, saying that I'm "sitting in an armchair" gives me the impression that I'm sitting comfortably rather than perched on one of the arms (much like you'd perch on a stool or a bed).
I'm thinking it might have to do with how many ways there are for someone to sit on a particular object: there's only one way to sit on a stool or on the average chair, but there are more possibilities on larger objects like armchairs, right? ^_^