Just making sure if a university has general education requirements, does it mean that everyone has to take those certain courses?

Yes, that's what "requirement" means. US universities operate on the principle that they should be preparing educated citizens-- people who know enough about a lot of things to make educated decisions. So that means that you have to take some science, some math, some English, a foreign language, etc. 

Only to an extent. 

The -entire- university doesn't have the same general education classes; they'll vary by school/college/faculty. I.e., the graduate medical school and law schools will have different requirements. The undergraduate engineering school will have different requirements than the college of arts and sciences, which will have different requirements than the business school. 

But, yes, the college's (or school's) general education requirements apply to everyone in that college. 

That doesn't necessarily mean that everyone has to take the exact same classes, though. For example, a college may have a foreign language as a general education requirement, but some people take Spanish to fulfill it, others French, German, Mandarin, etc. A college has a math requirement, but some people take statistics to fulfill it; others may take calculus. 

There was only one class that every single person in my college had to take: English composition. The rest of the general education requirements could be chosen from a list of several options. 

All regionally accredited universities have general education requirements, although they might all it something different like "core curriculum."  In some cases, the requirement is set in stone.  For example, all students must take English composition.  In other cases, there are options.  A student may be required to complete three lab science courses, but can pick among eight or more choices.