Design is generally the creation of a plan for the later creation of the proper project. In this design, it is stated how things are supposed to work, and when the project is created, these wishes are ideally to be fulfilled.
Game design is no different. In game design, you are establishing the Paida, Ludus and Narrative of a game. The Ludus is your overarching objective. When you complete a game, you are simultaneously completing the Ludus, as the completion of these is, in essence, the same thing. The Paida, meanwhile, is how the game is played. In Tetris, for example, you control falling blocks, and have to complete a row, which deletes these same blocks, giving you points.
Finally, the narrative. The narrative is a game’s story, and many argue it is one of the most important features of a game, thus sparking the “Ludology vs. Narratology” debate. A narrative can be found in all games, however, some feature it much more prominently than others. A notable example is Heavy Rain, a game which gives you little choice, and decides to instead play out very similarly to how a movie would.
It is these 3 elements which make a game. You want the player to be motivated, which is why games need a proper Ludus. You want players to have fun, which is why a good Paida is necessary. Finally, you want players to become more involved in the game, and reflt about it, become immersed, if you will. This is why the Narrative is important.
Game design also defines the tools of play, the components that define how a game is played. For videogames, it’d be a computer, a console, a phone or a tablet. However, in other games, you can see elements such as cards, a board and pieces and more.
Another element in games are the rules of it. In a videogame, these are defined by the Paida, but in all games, rules are what define things such as turns, rights and abilities of each player. One very small change to a rule may completely change a game.
In game design, it is the lead designer who is the main visionary and organizes the rest of the designers to complete the task. It is he who makes large decisions.
There also exists the game mechanics designer, or the systems designer. They are in charge of making the game balanced and take care of the Paida.
Finally, a there may be a level designer, who will be in charge of designing environements and game levels.
In conclusion, game design is all about projecting your idea of what a game will be, and making this very idea into what will be of the game.