Q. What is Computer Science exactly?
This is usually fundamental for any programmer, games or not.
Q. Wouldn't it be better to do an education specialized on Game Design?
A. No. You do that and you severely your career possibilities. If you do a Games related course and after two years you change your mind for whatever reason, switching careers to a non-games programming one is difficult because you don't have the underlying education that you would get from a Computer Science degree. However, to transition from a Computer Science degree to a Games Programmer career is much, MUCH easier.
Talk to your Careers adviser at College. He or she will give much more relevant information about your country's education system. Please bear in mind that the list I linked to are games companies
, not education.
Edit: Have a read of this article as well: http://scientificninja.com/advice/on-game-schools