Combination puzzles are puzzles which have pieces which can be moved in certain ways. There is a certain position of pieces considered "solved", and starting with a random combination, the goal is to get to that "solved" state.
Examples:
The Rubik's Cube is one of the first twisty puzzles to be created. It has 6 colors for its 6 sides, which must be joined together. However, it is best to think of the Rubik's Cube not by its sides, but its pieces. It has 12 edge pieces (which have 2 colors), 8 corner pieces (which have 3 colors), and 6 centers (which have 1 color) which don't move.
The 15 puzzle is one of the most popular sliding puzzles, containing 15 squares that can be slid around in a 4x4 square grid. These squares usually contain numbers, as shown above, however there are also variations that have images on them instead.