x86 refers to the general family of processors made by Intel, AMD, Cyrix, Siemens and etc. that use a common instruction set. Think of it as how we refer to certain human languages as being "latin based."
Within the x86 set of "languages" there are stair-stepped sub-languages. The most common three that you will run across in modern times are
For more information on the different CPU instruction sets, see the Wikipedia article x86 instruction listings and note how each chip introduced a new set of instructions that, generally speaking, improved upon and added to the previous set before it.
Once again, I speak in general terms when I say this: The newer the instruction set that the OS can "speak", the more advanced features within the chip that can be taken advantage of to the benefit of the OS.
i[n]86 refers to the highest set of instructions within the x86 family of processors that the program / OS will utilize. Specific generations of chips have specific subsets of instructions such as the 80386, 80486, 80686 etc. If an ISO is demarcated as being i386, it means that it doesn't use instruction sets for the CPU above the 80386 standard set. It will run on all x86 CPUs however.
Generally, the older the instruction set that a OS needs, the broader the possible hardware that you can install it on. However, i686 is broad enough these days to where you'll likely never have any trouble installing it unless you're trolling through the dumpsters behind a dollar store.
32-bit vs 64-bit: If an ISO is designated as 64-bit, it will not run on 32-bit hardware. It must have 64-bit hardware. If the ISO is 32-bit, it will run on 64-bit hardware, but not utilize the extra address space that the hardware can see. Think of the "bit-rate" (poorly chosen term, I know) as being backward compatible, but not forward compatible. You can fit 32-bits onto 64-bits, but not 64-bits onto 32-bits.
- 32-bit OS onto 32-bit hardware == Yep!
- 32-bit OS onto 64-bit hardware == Yep!
- 64-bit OS onto 32-bit hardware == Nope! (Chuck Tes... nevermind)
- 64-bit OS onto 64-bit hardware == Yep!