I have heard that different distributions of Linux have the same Linux kernel up to that the Linux kernels may be in different versions. Do different distributions of Unix also share the same kernel up to different versions? What are some examples of various distributions of Unix? I've only heard of BSD.
Linux is, in a narrow sense, a kernel. In a more common wider sense, “Linux” means a software distribution (an operating system and some applications) containing this kernel. See Is Linux a Unix?.
At any given point in time, each Linux distribution contains different versions of each of the thousands of pieces it's made of. They may additionally make slight modifications to some of the pieces, for example some distributions ship additional drivers with their kernels (usually, these drivers will make it into the official kernel after a few months — some distributions prefer to ship experimental drivers on the basis that partial support is better than no support).
There are other unix systems that are not based on a Linux kernel. See Evolution of Operating systems from Unix. The best-known unix-based operating system is Mac OS X; it contains a unix core (based on FreeBSD), but its GUI is different. Other than that, the BSD family is indeed the best-known “mainstream” unix other than Linux.
Unices other than Linux typically have a single official distribution. For example the FreeBSD project provides a kernel, a whole set of basic utilities, and a set of third-party packages. There are a few distributions in the Linux sense, that repackage software from other providers (e.g. Debian GNU/kFreeBSD, which ships a FreeBSD kernel, GNU utilities, and the software otherwise provided in Debian GNU/Linux), but they're rather marginal.