A game featuring a 1-2 players alternating mode generally has a single mode in which two people can play, but only one player at a time. Sometimes this will mean that (as is the case in Super Mario Bros.) you can select "2 players", but only one character can be controlled at any given time; once that character dies, then the next player tries with their character. This constantly bounces back and forth between the gamers in a single session. Another form of 2 player alternating is (as is the case in Animal Crossing, although that's 4 player alternating) when there are multiple "files", with one belonging to each player. A player can then choose their file, playing where they left off from their last save.
Some games also lend themselves easily to Hot seat play. This can be done outside of the game and works as long as a game has defining breaks in gameplay. For example, if one player dies he can hand the controller to another player whos "turn" begins where he continues using the next life and continues on until his life is used up. There does not need to be any game mechanic within the game to allow this.