This is regarding the Processor and their architecture.
i386 - is the old 386/486 CPUS
i586 - Pentuim/Pro and 100% AMD Compatibles like K5 or Cyrix
i686 - Started with the P II and then the P III and AMD Athlon, Duron, XP
To give a little more information on that, the X86 processors started with the Intel 8086 processor way back in 1978. They were incrementally improved (80186, 80286) and then Intel released the Intel 386 (i386) in 1980. That was then followed by he 486 (i486), the Pentium (i586), and the Pentium 3/4 (i686) and AMD's Athlon/Duron/T-bird (also i686).
Since all these processors were based on the same architecture (basically they read/wrote 1's and 0's in the same way), and their names all contained "86", the whole family was collectively called "X86". All the X86 processors were 32-bit.
The recent trend has been to move toward 64-bit processors, and several different architectures popped up. DEC's Alpha and Motorola's PPC chips have been 64-bit for a while, but Intel's Itanium and Xeon and AMD's Athlon64 are the new kids on the block.
The difference between the Itanium and PPC versus the Athlon64 is that the Itanium and PPC have completely different architectures (they speak different 1 and 0 languages), whereas the Athlon64 speaks the same language as the 32-bit X86 processors, but adds 64-bit registers. Therefore the name of the Athlon64 in generic terms is "X86_64".
Intel, not to be outdone, has since redesigned its 64-bit Xeon processors to use the same kind of architecture as the Athlon64, calling it "Intel 64-bit with Extended Memory Technoloty". Basically they couldn't say they copied AMD without being laughed at in irony (since AMD got its start by simply copying Intel's chips).