Game design is a process of: imagining a game; defining how it works; describing the elements that are going to be part of the game (concept, art, foundation); passing these data on to the team that will produce it.
And we can't forget to talk about game key-components . Game key-components are: core mechanics (the kernel of the game); the challenges and actions ; interface (how the player interacts with the game ); and storytelling engine (the immersive screenplay itself).
The graph below show us this idea:
We can use this ideas in any platform: board games, video games, war games, etc. It´s important to have a good sinergy between the player, the interface and the core mechanics.
If you like this concepts i'll suggest a very good book named Fundamentals of Game Design (ADAMS, Ernest. Fundamentals of Game Design. New Riders: 2009).