IA-32 ≈ x86

IA-32 (short for “Intel Architecture, 32-bit”, sometimes also called i386) is the 32-bit version of the x86 ISA. IA-32 is the first incarnation of x86 that supports 32-bit computing; as a result, the “IA-32” term may be used as a metonym to refer to all x86 versions that support 32-bit computing. certain iterations of IA-32 ISA are sometimes labeled i486, i586, and i686, referring to the instruction supersets offered by the 80486, the P5, and the P6 microarchitectures respectively

IA-64 ≠ x64

IA-64 is based on Itanium ISA, a server CPU architecture Intel created around 2000 INCOMPATIBLE with the “x86” processors in wide use on desktop systems. Itanium was supposed to be the “next best thing” but it proved not to bring the performance benefits that it was touted to have. It is on the way out.

x64 is a vendor-neutral/Microsoft term for 64-bit CPUs. A lot of Linux people will use the term “amd64” instead because AMD was the one that first came out with CPUs that extended the Intel “x86” 32-bit instruction set and architecture to 64 bits (rather than developing a new, incompatible 64-bit processor)