A game that is 1-4 players alternating has a multi-player mode that can support up to four players who alternate with one controller.
Typically, when a game offers alternating mode, there is usually a restriction on the gameplay that makes this mode the only possible playable solution. Golf games such as Mario Golf: Toadstool Tour are a great example of this; while consoles have enough processing power to allow split screen display, it makes games like Golf unplayable.
Sometimes, alternating play modes also allows the game designers to stretch the restrictions on the platform or the design of the game. Advance Wars: Dual Strike (via Hot Seat mode) and Animal Crossing are great examples of this.