I keep coming across the term "visible universe." Could this mean there is something beyond what we see or beyond space and time?
Yes, the term "visible universe" refers to the fact that the actual universe might be bigger than the part of it we can see (in fact, it might be infinitely big). The part we can see is determined by the age of the universe. For example, suppose that the universe is 13.7 billion years old, as indicated by recent measurements from the WMAP satellite. That means that the farthest away from the earth that we can see, in any direction, is 13.7 billion light-years - i.e., the distance light can travel in the time since the universe was formed.
Note that we are able to see objects which are currently farther away than this distance, since the universe is expanding, and the objects would have moved farther away in the time since they emitted the light which is just reaching us now. So what I really should say is that the visible universe contains all the objects whose light had to travel less than 13.7 billion light-years to reach us. For anything that is farther away, the light from it would not yet have had a chance to get here.