A schoolhouse is the building where a school lives — the house that is the school. People use it most for a small school in a village: often one building, one or two rooms, and all the children learning together. You will meet it a lot in American stories and in talk about the old days; the little red schoolhouse is a picture people keep of simple country schooling. In a big town today we usually just say school or school building. Schoolhouse sounds a bit old, or like the countryside. Though actually, it is only the building — not the lessons, not the teachers as a group, and not the idea of 'school' as a place you belong.