x86 is a generic term referring to the "instruction set" of the most successful CPU "architecture". The architecture is defined as how the CPU arranges and uses resources such as RAM and I/O. Intel invented the x86 architecture, and AMD, VIA, Cyrix and others have manufactured chips compatible with x86.
CPU's read bytes from RAM as instructions and do stuff based on that data, this is how programming ultimately works. All computer languages eventually have to be translated into this data, called machine language. A byte is a number 0-255, and the set of byte values and their corresponding instructions forms the instruction set. There are many different types of CPUs that adhere to a specific philosophy, and therefore each major type of CPU has its own instruction set that is not compatible with a different CPU. Other types of instruction sets include MIPS, PPC, ARM, 6502, Z80, 68000, and others. The two main types of philosophies are CISC (Complex Instruction Set Computing) and RISC (Reduced Instruction Set Computing). CISC packs more "power" into each instruction but requires the CPU to be slower and more complicated, RISC makes each instruction do less but makes the CPU cheaper and faster. x86 is a CISC CPU (as were most in the late 70's/early 80's when x86 was born) with RISC extensions. x86 as a name comes from Intel's numerical designation various chips in the architecture. It begins with the 8086, and then 80286, 80386, 80486, and onward through the Pentium chips, Core line and beyond (often called "686" chips even though they no longer have numbers).
To summarize, x86 is a term referring to the way that the CPU in the common PC was built and how it functions.
32-bit processors
on x86-64 processors, yes. On Itanium, no.
The term refers to hybrid processors
Actually, they both make what are known as x86-64 processors, which are processors that can run both 32-bit and 64-bit programs. Check the specifications of the processor for something like "64-bit" or "64-bit Ready" to make sure you can run both x86 (32-bit) or x64 (64-bit) programs!
IBM AIX currently only supports Power chip processors, which are not the Intel X86 format.
x86
Any x86 processor from the 386 on up to present-day processors can be used with Windows 98.
Planned x86 processors will have the SSE4 instruction set.
All of them. All drivers for video cards running on x86 processors are compatible with Celeron processors.
Yes, Apple's Intel transition was the process of changing the CPU of Macintosh computers from PowerPC processors to Intel x86 processors.
Only if the processor has special 64-bit extensions, known as AMD64 or EM64T. Certain Pentium 4 processors, all Core 2 processors, and all AMD Athlon 64 processors have these extensions, and can thus run 64-bit operating systems and programs.
There aren't really kinds of processors, they have different architectures. There are thousands with the most popular kind of architecture in processors are x86/x64 and ARM.