The relationship that I see most often is that universities are made up of colleges. As in, a college usually addresses a (reasonably) related topic and only grants degrees on that topic/area. A college has a dean that heads it. Once a college grows large enough to split into two colleges (two deans), then you get a chancellor who oversees both colleges and now you've got a university. You can then add more colleges as you see fit, and it's still a university just with more colleges and deans.

The term "community college" is a US term that speaks to an educational institution that usually focuses mostly on associates (2 year) degrees as a post K-12 school. Community colleges can do four year degrees, but if they get too much of them, they will change their name to just be a college or university at that point.

Yes, there are schools that call themselves universities in US even though they only have what amounts to a single area of degrees and should be called a college just because the title "university" is more prestigious. No, there's really no governing bodies that determine what can be called a college or university, so it's really just the name of the corporation. All of this is more like guidelines ala the Pirate Code.