AMD64 refers to the x86-64 (64-bit) instruction set. It can be confusing because it goes by different names: amd64, intel64, x64. etc. but they all refer to the same thing, the AMD implementation of the 64-bit x86 instruction set that has become the de facto standard for 64-bit platforms. Intel had it's own 64-bit architecture, called Itanium (IA-64), but it has largely been abandoned. The most significant advantage to x86-64 was that it implemented the complete 16-bit and 32-bit instruction sets, along with the new 64-bit support, which meant that it was backwards compatible and could run 16-bit and 32-bit code without performance penalties. IA-64 provided x86 support through emulation, and therefore was slower. IIRC, AMD and Intel basically did a cross-licensing "patent swap" for various technologies each company owned, ensuring that x64 processors from both companies would be completely compatible with one another.
More than you wanted to know, I'm sure, but there you have it.
Edit: And apparently I'm not spelling very well this morning, either.