chess problems, are known as fairy pieces. For a list, see fairy chess piece.
Physical chess pieces used to play a game are usually three-dimensional figurines, taller than they are wide (a set of pieces designed for a board with squares two inches wide will typically have a king around 3.75 inches tall). They are available in a variety of designs, with the most usual known as the "Staunton design", named after the 19th century English chess player, Howard Staunton, being designed by Nathaniel Cook.
For games played at the top level, pieces made of wood are usual, but for lower-level games or very large tournaments, plastic sets are more normal.
Some small magnetic sets, designed to be compact, have pieces more like those used in Shogi and Xiangqi - each piece being a similar flat token, with a symbol drawn on it to show which piece it is.
See Also: Lewis chessmen, Charlemagne chessmen, Rules of chess.