At it's core level, a game is just a puzzle that is (usually) too difficult to completely solve, because of the complexity and ambiguity in the mechanics, such as randomness, player interaction, and other hidden information.
Types of puzzles include:
Spatial Navigation (such as sliding or moving blocks, or navigating a maze)
Common Sense (using simple outside knowledge to solve a problem)
Logic Puzzles (using mathematics and deduction to find a solution)