What is the difference between these if any. Will they all run a computer with a AMD Sempron 64 bit prossesor with 160gb hard drive and 512mb ram?

Yes, the x86_64 is the 64bit version. I am running Slackware 10.2(32 bit OS) on my AMD64 laptop. I have had no problems as long as I used the test26.s kernel when installing.

The rest are 32 bit processor types.

i386 --> means it's designed specifically for basic intel architecture based on the 80386 (or 386 computers)
i486 --> goes up to architecture for 80486 (486 and above computers). NB: 386s might have problems with some of this
i586 --> as above but designed for Pentiums and above. Pre-Pentiums may have problems.
i686 --> Pentium II and above.

The 386, 486, 586, and 686 are/were Intel x86 (32 bit) processors. The x86_64 is AMD's (and Intel's) 64 bit extension to the x86 line that is backwards compatible with previous x86 chips. All will run on an x86_64 processor like your Sempron, however the non x86_64 packages won't take advantage of the 64 bit features of the platform. Likewise, the i386 binaries won't have any 486 or later specific optimizations/improvements, and so on.

i686 --> Pentium Pro and above

Yep, made a mistake there.

The x in x86 comes from the fact that x is generally used as a "variable" and consequently, x can be 3,4,5 or 6. Intel's processor have been named 386, 486, Pentium, Pentium II and so on.

The fact that they are all backward compatible implies you can install a 386 distro on a 686, but not the other way around.

Also regarding 64 bit processors intel came up with a 64 bit architecture but it is totally not compatible with 32 bit processors, which means you simply can not run any os designed for 32 bit machines on these machines. I think they are called IA-64 (I believe if you try to run a 32 bit instruction set on a 64 bit process, there is a severe performance penalty.

However, AMD came up with an architecture that was 64 bit, but was essentially an extension of 32 bit. This was called x86-64, which means it can run 32 bit instructions just fine. Therefore, if you have a 64 bit processor (AMD Sempron, Athlon, Turion etc) it can run a i386 OS as well. Intel called their technology EM64t (Extended Memory 64 Technology).

Actually if you think about it, x86 itself is more accurately x86-32, since it is an extension of the 16 bit processor era of 8086 !