After the x86 32-bit processors, Intel wanted to move to 64-bit, and co-operated with HP to develop the 64-bit Itanium processor. Unfortunately, this was not very populare, since the instruction set and the architecture was rather different from x86.
AMD jumped in and extended the know x86 architecture to 64-bit, first calling it EM64T, but then AMD64. Just like AMD had to follow Intel with 32-bit processors, Intel now had to follow AMD with the 64-bit processors, since AMD's 64-bit architecture proved to be much more populare than the Itanium 64-bit processor.
Of course, Intel doesn't like it to be called AMD64, that's why they call it x64, but essentially AMD64 and Intel's x64 are compatible (except for some minor differences, see http://en.wikipedia.org/wiki/X86-64#Differences_between_AMD64_and_Intel_64). If you compile for x64 or AMD64, the generated code will avoid those differences so in practice you can run on both.
In any case, try to avoid generating for Itanium (also called IA-64) because this is a totally different kind of processor.