Traditionally, chess puzzles have been published in newspapers, and you'll even find compilations of them in books. Today, Web sites offer chess puzzles for the casual beginner, intermediate or advanced player to stretch his or her mental muscles.
The most common puzzles, directmates, ask an individual to win in a certain number of moves. Two-movers require you to win in two moves, three-movers require a win in three moves, and those that require more moves are called more-movers. A well-composed puzzle of this kind will only have one possible solution in the expressed number of moves. It will also be a foolproof solution that assumes the other side will play its best defense.
A puzzle will usually have you play as the white side, which customarily plays from bottom to top on a two-dimensional page. This also makes it easier to read solutions written in algebraic chess notation, which is a system of describing chess moves in letters and numbers and is interpreted easily from white's perspective.
A chess problem is a particular kind of directmate puzzle that is unlikely to occur in a real game, but rather composed in order to be difficult, original or even beautiful in its theme. And while problems require players to win within a certain number of moves, studies are yet another kind of chess puzzle that asks a player to win or draw, but with no restrictions on the number of moves.
If you like the idea of chess problems for their own sake and not necessarily to improve your playing skills, you might also try heterodox puzzles (also known as fairy chess). Consider the selfmate, where white tries to force black to deliver checkmate in a certain number of moves, while black resists. The helpmate is similar, but black cooperates with white. A reflexmate is a selfmate with the additional rule that if either side can deliver checkmate, it must.
But puzzles can get even odder. The selfmate being four centuries old, some consider it downright orthodox. Today, fairy chess more commonly refers to puzzles with stranger conditions or even unorthodox pieces (the nightrider or the grasshopper are common examples). With endless possibilities, chess puzzles themselves can easily become a lifelong study.