Is there a difference between a (under)graduate and honours studies - also concerning thesis? Is there a difference between an undergraduate and an honours thesis? I came across the latter during some online research on US colleges and it seems to me that honours are only known in those institutions. Is this correct?

Moreover, it seems that some colleges offer normal undergraduate as well as undergraduate honours studies/classes/degrees. I heard that the difference is that the latter have to write a thesis while the former seemingly don't. Is this correct and are there other differences?

Does this separation also exist on the Master's level and graduate schools?
Just about every possibility exists somewhere in the United States. There is no one answer. 

Some colleges or universities do not require any type of thesis for an undergraduate degree. Others may require students in an honors program only to complete a thesis or project. Yet others may require all students to complete a thesis or project for an undergraduate degree. 

At the graduate level, there are research master's degrees and non-thesis/course-only master's degrees. The latter may still have a project or paper. A research master's degree is usually for those who are going on to a Ph.D.