LLVM originally stood for "low-level virtual machine", though it now just stands for itself as it has grown to be something other than a traditional virtual machine. It is a set of libraries and tools, as well as a standardized intermediate representation, that can be used to help build compilers and just-in-time compilers. It cannot compile anything other than its own intermediate representation on its own; it needs a language-specific frontend in order to do so. If people just refer to LLVM, they probably mean just the low-level library and tools. Some people might refer to Clang or llvm-gcc incorrectly as "LLVM", which may cause some confusion.
llvm-gcc is a modified version of GCC, which uses LLVM as its backend instead of GCC's own. It is now deprecated, in favor of DragonEgg, which uses GCC's new plugin system to do the same thing without forking GCC.
Clang is a whole new C/C++/Objective-C compiler, which uses its own frontend, and LLVM as the backend. The advantages it provides are better error messages, faster compile time, and an easier way for other tools to hook into the compilation process (like the LLDB debugger and Clang static analyzer). It's also reasonably modular, and so can be used as a library for other software that needs to analyze C, C++, or Objective-C code.
Each of these approaches (plain GCC, GCC + LLVM, and Clang) have their advantages and disadvantages. The last few sets of benchmarks I've seen showed GCC to produce slightly faster code in most test cases (though LLVM had a slight edge in a few), while LLVM and Clang gave significantly better compile times. GCC and the GCC/LLVM combos have the advantage that a lot more code has been tested and works on the GCC flavor of C; there are some compiler specific extensions that only GCC has, and some places where the standard allows the implementation to vary but code depends on one particular implementation. It is a lot more likely if you get a large amount of legacy C code that it will work in GCC than that it will work in Clang, though this is improving over time.