Game design focuses mostly on the actual gameplay, what NPC the player encounters where, how much damage a weapon does, etc. There can be storyline work, writing dialogue or scripts for cutscenes, etc. There’s typically also a lot of math involved. (The actual art for the characters and cut scenes is done by artists, yet another separate job title, usually after getting direction from the design team.)

If you’re interested in working as a game designer, it doesn’t really matter what college you go to or what you major in. Pick a good 4 year college near an industry hub like Los Angeles, San Francisco, Austin, Seattle, etc. Pick a major that you enjoy and that will allow you to graduate as quickly as possible — in the long run, time spent working in the industry will matter a lot more than time spent finishing a specific major.

Along with whatever you decide to major in, take math through Calculus 1, at least two courses in Statistics, two courses in writing, one art course, and one programming (Computer Science 101) course. Fill in the rest of your course requirements with a wide variety of classes: sociology, psychology, history, literature, sciences, etc, will all be useful to you at one point or another in your game design career. As far as the art and programming classes, you don’t have to be particularly good in either one to be a good designer, but taking even just the one class in each will help you talk to your colleagues down the line.