The rental of apartments or private flats in Prague is more like a newer trend, appreciated especially by families with children and people in search for a cosy, well equipped place (kitchen, bathroom, TV set) where they can do almost everything by themselves.
Apartments can be owned by an agency that deals with the tourists (clients) or can be owned by a person that offers it for rent. Private owned flats can sometimes be better equipped than normal apartments. Incoming tourists get their key of the flat or apartment and can use the place as if it were their home during the days of their stay.
This type of accommodation has the advantage of being cheaper than a three star hotel room, but on the second hand, there is no reception or room service to deal with your tourist needs.