Hard to explain what specifically game design is. In my views its the aims and objects that help set down what the design is. take for example Space Invaders. Everyone has played space invaders, such a simple and original game. To be honest does anyone actually need to be taught how to play it...no. You just stick someone infront of the screen and they instantly know how to play it. The aim of the game is to shoot down all the little invaders without getting hit and before the time limit runs out.
Every game has the same basic quality as space invaders. Every game has aims and objectives to complete, otherwise what would be the point of the game, it wouldn't be a game.
Another part that helps to explain game design could be the environment, the colour, the realism. Making something realistic helps the character to believe in what they are doing. Games are not like space invaders anymore. Games have evolved so much that some people become part of the game, such as online role playing. Not so much the game being realistic as such but it has to be believable.