x86
means Intel 80x86 compatible. This used to include the 8086, a 16-bit only processor. Nowadays it roughly means any CPU with a 32-bit Intel compatible instruction set (usually anything from Pentium onwards). Never read x32
being used.
x64
means a CPU that is x86
compatible but has a 64-bit mode as well (most often the 64-bit instruction set as introduced by AMD is meant; Intel's idea of a 64-bit mode was totally stupid and luckily Intel admitted that and is now using AMDs variant).
So most of the time you can simplify it this way: x86
is Intel compatible in 32-bit mode, x64
is Intel compatible in 64-bit mode.