As Cap says, sofa and couch are pretty well interchangeable here and so is settee. The OED describes a couch as having a half back with one arm at that end only, but general usage is as for sofa and has been for as long as I can remember though that could have been an import from Hollywood.

Donít know about davenport, but a chesterfield is a subset of sofa. The chesterfield sofa is a type of upholstered settee that first appeared in England in the early 19th century. The first fully upholstered and sprung sofa, it may have been named after the town in Derbyshire. More likely, it is named after the 6th Earl of Chesterfield who was something of a fop; a type of double breasted overcoat is also named after him. The chesterfield has upright (as opposed to sloping) rolled arms that come up to the same height as the back of the sofa. Chesterfields are often upholstered in leather, with buttoned backs, but fabric upholstery is also common.

It looks like this: