You want to give the player a reason to keep playing the game, by introducing some variation into the game, or you want a mechanism by which to transparently ratchet up the difficulty of the game.
A common way to introduce variation into the game is to add a level system. Each level is usually a discrete map, with a different layout and different obstacles. However, levels can also use the same map by changing the difficulty level (e.g. giving the player less health, etc.). Generally, the player will need to finish one level to move onto the next level.
Levels can also be useful as a mechanism to communicate increasing challenge to the player. Each successive level provides the game designer an opportunity to raise the difficulty of the game; the player can easily tell that he is in for a more difficult experience as he moves up in levels. For example, in Tetris the blocks fall much faster in level 50 than in level 10.