Elements of game design, part 1.

"Game design" is really a vague term. It incorporates many things. As Wikipedia states, it is a stage of production, where the foundations of setting, gameplay, visual and audio style are created and refurbished. Some people also call the further creation of a game "game design", but again - the term is perceived differently by different people, that's why it is so unspecific. I'd probably agree with those who think that this phrase can describe the whole process of creation of the product, but also agree that the most important part is done by the team at the beginning. Afterwards artists, animators, programmers mostly have to follow the rules the team has set in the first phase.
Aw, well. There sure is some magic in the creative process and if I try to explain it to myself, the effect will be gone. All right.
People always play. Johan Huizinga even wrote a book about this phenomena, "Homo Ludens" (Playing Man). In it he goes about showing the way people incorporate play in their everyday life - in communication, relationships, private lives. I think that video games at first were just digital games, but now they are mostly a cross-breed of play and books, movies, plays. Another way to create a new layer of reality on top of the common one we live in. Some people blame those who get emerged in those fake stories too deeply. But is it different from reading too much books or watching films? The reality we sense is just as "fake" as the reality of Half-Life series. That is to say that our brain assembles a collage of outer world using the information that streams in through our sensory systems, and does almost the same thing when a preson reads a book or plays a game. But I am getting distracted.

Anyways, what is game design all about? Creating a logical, consistent world that a player can immerse himself into. The world can be as small as a street in Frogger, or as big as Azeroth in Warcraft franchise, size doesn't matter. It is all about how believable it is. Then, you have to create adequate play mechanics, i.e. - gameplay. Player can't do random, unrelated actions throughout the game. Set of his action must be limited, but wide enough to still be interesting. FPS - walk around and shoot, RPG - walk around, collect loot and spend points at talent tree, RTS - command units on a map. The genre is defined by this set of actions player perform, distinguished by type of gameplay. 
So, yes. Games in a certain sense are just like people. Similar in general, but different and unique in details. Unless we are talking about the stuff EA, Activision or Nintendo releases, of course.