Any 64 bit processor can run amd64, this means any amd 64 bit processor, or any new-ish intel, anything celeron D or later, and all the core 2's, and some others. That said, 64 bit processors can also run 32 bit (x86) versions as well. If you arn't sure, you can just use the 32 bit edition.
The average user will never notice the difference (unless you need extreme floating point precision?).
The difference between the two is the set of instructions that can be fed to the cpu. the 64 bit processors/os's have additional instructions that allow for certain things, when you run a 32 bit os, you just don't use those instructions, which in normal use, is just fine, unless you do huge scientific number crunching
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