The answer is complicated. Depending on the studio size, number of people, and task specialization, game developers do a lot from day to day. (I'm going to assume you mean game developer instead of just game designer, since developer is the broader term, and encompasses just about anyone who puts work into making a game, whereas designers are the ones that effectively tie the work together to make the finished product, or who come up with arrangements of elements and game mechanics to make the overall experience.)
In a smaller studio, game developers wear many hats. On any given day, one could be making art assets, recording or making sounds, writing code, creating levels, producing or proofreading story, or doing housekeeping or production activities, which could be anything from taking out the trash to burning CDs to doing accounting or marketing work. Small studio models produce people who are broad generalists; because the tasks are so plentiful compared to the number of workers, a smaller game studio requires that each person be able to contribute multiple skills to the game, and that they balance their work efforts so as to not neglect something that is needed for forward progress.
This all goes out the window when looking at large game studios. At a large game studio, there are usually more people than tasks (keeping to realistic development schedules), so subject fields (art, sound, programming, modeling, animation, game design, level creation, writing, business, and so on) typically each have their own divisions within the studio, and these are sometimes broken up even further based on individual skills or subsets of related knowledge (game programming vs. engine/tool programming, or character modeling vs. weapon modeling vs. environment modeling). In large studios, the type of person that is produced is a narrow specialist, or (in better studios) a "T-shaped" specialist. These people become very knowledgeable and productive in one or a few closely-related and specialized areas of expertise, and are also called subject matter experts.
Now, if you want a job description of a game designer, rather than a game developer, the game designer is the person whose job is to come up with the general design of a game, selecting and tying together features or aspects which will lend toward the intended experience delivered to the player, planning and distributing content creation tasks, and taking produced content and tying it together into a game. If they don't share responsibilities of content creation of their own (usually, they can double as programmers, level designers, artists, or writers), then they typically use a high-level tool (either a third-party one, like Unity or Unreal, or an in-house one for custom engine work) to bring together and manage the content of the game. They also generally produce documentation about what features and style the game should have, and ensure that everyone on the team has a shared vision of what the final product should look and feel like.
I hope this helps; I've learned all of this from a combination of online resources and developers' conferences, as I haven't gotten to produce any full games of my own just yet, outside of school projects. If you want more reliable sources, look for the proceedings for game developer's conferences and summits, and that should give you a good place to start. Remember to cite your actual sources!