An endgame study, or just study, is a composed position—that is, one that has been made up rather than one from an actual game—presented as a sort of puzzle, in which the aim of the solver is to find a way for one side (usually White) to win or draw, as stipulated, against any moves the other side plays.
Composed studies predate the modern form of chess. Shatranj studies exist in manuscripts from the 9th century, and the earliest treatises on modern chess by the likes of Luis Ramirez Lucena and Pedro Damiano (late 15th and early 16th century) also include studies. However, these studies often include superfluous pieces, added to make the position look more "game-like", but which take no part in the actual solution (something that is never done in the modern study). Various names were given to these positions (Damiano, for example, called them "subtleties"); the first book which called them "studies" appears to be Chess Studies, an 1851 publication by Josef Kling and Bernhard Horwitz, which is sometimes also regarded as the starting point for the modern endgame study. The form is considered to have been raised to an art in the late 19th century, with A. A. Troitsky and Henri Rinck particularly important in this respect.
Most composers, including Troitsky, Rinck, and other famous figures such as Genrikh Kasparyan, are known primarily for their studies, being little known as players. However, some famous players have also composed endgame studies, with Emanuel Lasker, Richard Réti, Vasily Smyslov, and Jan Timman being perhaps the most notable ones.
Richard Réti's study is one of the most famous of all time. It is White to play and draw (see Réti endgame study). At first sight, this seems an impossible task: if White tries to chase after Black's pawn he can never catch it (1. Kh7 h4 2.Kh6 h3 etc. is clearly hopeless), while it is clear that Black will simply take White's pawn if he tries to promote it.