GNU isn't really an OS. It's more of a set of rules or philosophies that govern free software, that at the same time gave birth to a bunch of tools while trying to create an OS. So GNU tools are basically open versions of tools that already existed, but were reimplemented to conform to principals of open software. GNU/Linux is a mesh of those tools and the Linux kernel to form a complete OS, but there are other GNUs, e.g. GNU/Hurd.
Unix and BSD are "older" implementations of POSIX that are various levels of "closed source". Unix is usually totally closed source, but there are as many flavors of Unix as there are Linux (if not more). BSD is not usually considered "open", but it was considered to be very open when it was released. Its licensing also allowed for commercial use with far fewer restrictions than the more "open" licenses of the time allowed.
Linux is the newest of the four. Strictly speaking, it's "just a kernel"; however, in general, it's thought of as a full OS when combined with GNU Tools and several other core components.
The main governing differences between these are their ideals. Unix, Linux, and BSD have different ideals that they implement. They are all POSIX, and are all basically interchangeable. They do solve some of the same problems in different ways. So other then ideals and how they choose to implement POSIX standards, there is little difference.
For more info I suggest your read a brief article on the creation of GNU, OSS, Linux, BSD, and UNIX. They will be slanted towards their individual ideas, but those articles should give you a better idea of the differences.