Puzzle games is 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, as with the popular game Tetris.
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 most games include elements of puzzle games. Action games include puzzles every once in a while to break up the fighting, like God of War. Platformers like Prince of Persia: The Sands of Time make puzzles out of figuring out how to get from one part of the room to another. Adventure or RPG games like The Legend of Zelda incorporate what could just be a piece of a bigger puzzle game as an obstacle in a dungeon. sometimes if a player gets a puzzle wrong it may activate a trap.