When the term 'x86' is used, does it always refer to 32-bit? And is it only about Intel processors?
x86 is shorthand for "The family of 80386 and upwards processors" where "x" stands for the number 2 and upwards. (There was an 80186 processor but that was only used for a special case, and the 80286 was not much more than a glorified 80186). All of them were based on the 8086, a very popular 8/16 bit processor of the 80's (8/16 means it was an 8 bit processor with some enhancements that made it almost like 16 bits).
The 80386 (known just as the "386") was 32 bit. The 80486 (known just as the "486") was 32 bit. The 80586 (the first generation of Pentium processors) was also 32 bit. Processors in the same family since the 586 have been either 32 bit or 64 bit.
x86 also refers to the instruction set these processors use. All the processors in this family use the same core instruction set, but with a few differences:
AMD have their own notation for similar processors, but they still speak the x86 language at the core. amd64 provides similar functions to x86_64, and AMD's 3dNow! provides similar facilities as MMX and SSE from Intel.
Intel also made another architecture - the IA64, or iTanium. This uses a completely different instruction set, and is only used in very high-end machines - mainframes, industrial systems, etc.
So in short "x86" is taken to mean "Any processor which, at its core, uses the Intel 32-bit instruction set based on the original 8086 instruction set" - be that Intel, AMD, Via, or others.