i586 is about the 5th generation of the intel 86 architecture
It goes something like this,
8086 and 8088, 1st gen
80286, 2nd gen
80386, 3rd gen (this is the minimum for linux)
80486, 4th gen
80586, 5th gen (intel pentium and amd k5)
80686, 6th gen (intel pentium pro, II and amd k6)
after that it starts getting a bit strange, if they kept the same system i belive it would be something like this.
80786, 7th gen (intel pentium III and 4. amd athlon(plain and XP), duron and sempron)
80886, 8th gen (later gen intel pentium 4. and amd opteron and athlon64)
But really after 80686's intel and amd followed there own generation schemes.
There is also older makes like winchip and cyrix, and newer ones like viaC3's (which i think originated from the cyrix stable) to throw into the mix.
In general anything that is compiled for a 386 will run on anything above it. However for an example something compiled on a 486 will not run right on a 386.
When you talk about mac's they used a chip that originated from motorola named the PPC, which from what i understand is now made by ibm and i think is also the same chip that powers some sparc (please correct me if i'm wrong). But the latest generation of mac's come with an intel chip (the core solo or duo) which is based if not the same as the pc chip.