Puzzle games is a game genre that mainly focuses on the thinking skills instead of quick reflexes of the player. Puzzles games are those where you need to solve a puzzle (e.g. a jigsaw); however, action puzzle games are also included, such as falling blocks that must be arranged into either lines or patterns, such as with the homebrew game of Vectris, which can be found on the All Good Things cartridge, as well as Colorclash.
Most puzzle games don't follow a story. They are often just graphics of blocks or icons or whatever the game uses. There are rarely any characters or bosses that you'd find in other games. A high score is often a goal.
It should be noted that several games include elements of puzzle games (even if they are not entirely a puzzle game per se), such as with Spike Hoppin'.