The schooner is not a specific kind of vessel, but a large family of ship types. What every schooner has in common is that it's mainly fore-and-aft rigged (the main sail must be a fore-and-aft sail), and that it has at least two masts (a foremast and a mainmast; a two-masted vessel with a mainmast and a mizzen mast is not a schooner).
A Fore-and-aft schooner is rigged solely with fore-and-aft sails. Two-masted fore-and-aft schooners are small, fast and handy vessels, often used as fishing vessels in the 19th century. Three-masted fore-and-aft schooners were at one time very common for trading in the Baltic. There have been large four-, five-, and six-masted fore-and-aft schooners as well, especially in America, but their gaff sails were so large that they became very hard and dangerous to handle. There was even one seven-masted fore-and-aft schooner, the only seven-masted sailing vessel ever, the Thomas W. Lawson, built to carry coal. She was considered a man-killer; for huge sailing ships like these square sails are much safer.
A Topsail schooner is similar to a fore-and-aft schooner, but carries square topsails, and sometimes a topgallant sail on its foremast. A few topsail schooners were rigged with square sails on two of their masts; such a vessel is called a two-topsail schooner.
Barquentines, brigantines and hermaphrodite brigs can be said to also belong to the family of schooners; indeed, in Scandinavian languages they are referred to as such.