This simple and popular memory game is also known as Concentration or Memory. In Japan it is called Shinkei-suijaku. In Britain it is called Pelmanism after Christopher Louis Pelman, who founded the "Pelman Institute for the Scientific Development of Mind, Memory and Personality" in London in 1899.
Two or more people can play. A standard Anglo-American 52-card pack can be used, or any other kind of cards, provided that the backs are indistinguishable and the faces can be matched up in pairs or quartets.
The cards are thoroughly mixed and spread face down on the playing surface - table or floor. The cards should not overlap, and can be arranged in a regular pattern or spread irregularly as desired.
Players take turns to play. At your turn you turn face up two cards of your choice from the layout. If they match, you take these two cards, store them in front of you, and take another turn. If they do not match, you turn them face down, without changing their position in the layout, and it is the next player's turn.
When playing with an ordinary 52-card pack, cards "match" if they have the same denomination - a six matches a six, a king matches a king and so on. Suits are ignored.
Play continues until all the cards from the layout have been taken as matched pairs.
Each player scores one point for each pair taken. The player with most points is the winner - if two or more tie for most points they all win.
Some special packs for this game have only two identical cards of each type, making it harder to match cards at the beginning. Since the game in any case becomes easier the fewer cards that remain in the layout, it seems preferable to reduce this effect by having four of each card, so that the first card of a new type has three possible pairings rather than just one. But those who prefer to play with just two cards of each type using a standard pack can agree that cards only match if they are of the same denomination and the same colour.
The Wikipedia article on Concentration explains several variants of this game.
The collection HOYLE Card Games for Windows or Mac OS X includes "Memory Match", a version of Pelmanism featuring wild cards and bonuses, which can be played Solitaire or by two players, along with many other popular card games.
Gabor Kertes has published a Concentration - the Memory Games for Windows, in which several versions of this game can be played solo against the clock, or against computer or human opponents.
At VanillaGrape you can play Memory, an online Pelmanism game against the computer.