The x86 term became common after the introduction of the 80386, it usually implies a binary compatibility with the 32-bit instruction set of the 80386. This may sometimes be emphasized as x86-32 to distinguish it either from the original 16-bit x86-16 or from the newer 64-bit x86-64 (also called x64). Although most x86-processors used in personal computers and servers have 64-bit capabilities, to avoid compatibility problems with older computers or systems, the term x86-64 is often used to denote 64-bit software, with the term x86 implying only 32-bit.
A non-Intel historical perspective: The follow-ons to the Intel 8086 processor included the 80186, 80286, 80386, and 80486. It was convenient to refer to the common underlying architecture as 80x86, which was then abbreviated to "x86". This is commonly used, but is not particularly precise since it includes both 16-bit processors (80186 & 80286) and 32-bit processors (80386 & 80486). Still, it has proven a useful shorthand since few people care about the older 16-bit processors.
Intel's official name for the 32-bit architecture starting with the 80386 processors is "IA-32", and is still used to cover the 32-bit mode of operation of the current chips. Clearly, "IA-32" is not a good name for the 64-bit processors that came more recently, but "IA-64" was already in use for the architecture of the Itanium processor family. AMD (under license from Intel) developed a 64-bit architecture by extending the IA-32 architecture, calling this "AMD64" -- also a name that would not work well for Intel! The AMD64 architecture was also referred to as the "x86-64" (based on the "x86" abbreviation discussed above) or sometimes "x64" architecture. Intel initially called their 64-bit implementation "EM64T", but later changed to referring to this as the "Intel 64" architecture.
Intel's software developer's manuals mention both "Intel 64" and "IA-32" in the titles, as most Intel processors continue to support both architectures.