There are two competing definitions of Asynchronous Gameplay:
1) Asynchronous gameplay, as defined by Nintendo, is multiplayer gaming in which players are experiencing the same game very differently. It is a central feature of the Wii U console, where one player can use a Wii U gamepad while playing with or against players using the Wii remote. For example, in the mini-game "Luigi's Ghost Mansion" from Nintendo Land, the player using the gamepad is a ghost who can see both himself and the players on the gamepad's touchscreen, while the Wii remote players cannot see the ghost's location on the TV screen.
This style of asynchronous gameplay predates the Wii U. The 2003 online multi-player PC game Savage: The Battle for Newerth also relied on asynchronicity; players formed two armies, with one player on each side taking a commander role and playing a birds-eye-view strategy game while the rest were soldiers playing an action game.
2) The Nintendo definition is predated by a radically different one referring to games in which players take turns. While this could include something simple like a game of checkers, it could also be a strategy game in which each player submitted a series of movies and nothing happened until all players had locked in their game. In this definition, asynchronous gaming means that one player can engage in the game while the other ignores it to do something else until their turn comes.