What is i386 and i586 and i686

These are different generations of Intel 86 architecture. 
8086 and 8088, 1st gen
80286, 2nd gen
80386, 3rd gen 
80486, 4th gen
80586, 5th gen (intel pentium and amd k5)
80686, 6th gen (intel pentium pro, II and amd k6)

I386:


Intel 80386 was a 32-bit computer processor more commonly known as the i386 or the 386. The i386 was backward compatible with hardware that targeted earlier x86 chips. The i386 was also the first consumer processor to support 4 gigabytes of memory. The low price of the i386 made possible the growth of the personal computer market


I586:


Most commonly seen as a suffix to binary packages (such as RPM packages) to be installed on a Linux system. It simply means that the package was designed to be installed on 586 based machines, ie. 586 class machines such as the 586 Pentium-100. Packages for this class of machine will run on later x86 based systems but there is no guarantee that they will run on i386 class machines.


I686:
It simply means that the package was designed to be installed on the 686 based machines, ie. 686 class machines such as the Celeron 766. Packages for this class of machine will run on later x86 based systems