x64 is used as a short term for the 64 bit extensions of the "classical" x86 architecture; almost any "normal" PC produced in the last years have a processor based on such architecture.
AMD invented the AMD64 extensions; Intel was more or less forced to implement them, and called them first IA-32e, then EM64T and finally Intel 64 (actually, the AMD and Intel extensions aren't exactly the same, but they are almost identical).
Many people also call this stuff x86-64, to have a vendor-independent name and to stress the fact that it's the 64 bit evolution of the x86 architecture. All the "regular" PCs that are sold with "64 bit processors" run on x86-64 architecture.
IA-64 (Intel Architecture 64) is an almost completely unrelated 64 bit architecture (also known as Itanium), developed by Intel initially for high-end servers. It was said that Itanium could have been a replacement for the x86 architecture, but this architecture didn't have much success (for various reasons), so it's unlikely that you'll ever need the IA-64 installers.
For more information, you may have a look at the wikipedia articles on x86-64 and Itanium.
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