A starship class is the basic design of a starship. In most science fiction universes, a ship's class will generally be named after its first ship, known as the class leader or pathfinder.
Classes of starships, like their terrestrial counterpart, are used to save effort and money - it's considerably easier and cheaper to build multiple units to the same set of plans than to develop a new unique plan for every individual starship.
† indicates a conjectural or non-canon class.