They may all have exactly the same meaning, they do not all have the same context.
Cuarto - Quarters. Living quarters. It's a very ancient term in English, but it still lingers in the spanish language.
Recámara, to me, makes me think of a hotel room, or a room for rent.
Alcoba, I have never heard, but judging from the others' attempt to translate it, I find it to mean nothing more than a
room. Not a bedroom in particular, but rather just a room.
Aula is something I know has classroom.
Habituación sounds like a very proper, fancy way to refer to a bedroom; I can see why it would be used in Spain.
Dormitorio is exactly what you would think it is, a dormitory. In English, the word
domicile are very proper names for a bedroom.
So again, even though they all mean the same thing, they deserve their contexts.