The 32-bit Intel processors are known as x86, and the 64-bit Intel processors are known as x86–64.
The explanation for this is historical, but Intel’s iconic processor line began with a model known as the 8086. Later versions were referred to as 80×86 processors, or x86 for short, and had designations like 80286 and 80386. The x86 architecture changed over time, but the core instruction set stayed the same, enabling software written for earlier x86 processors to run on them.
AMD, an Intel rival, eventually began producing processors with the same instruction sets as Intel processors, allowing them to run the same software and support the same peripheral hardware. As companies like Sun and DEC started shipping 64-bit processors for their RISC platforms, AMD built a 64-bit chip based on the x86 instruction set and architecture that simply added 64-bit instructions while leaving the 32-bit ones in place, enabling it to run both 32-bit and 64-bit programs. For a while, this was referred to as the AMD64 family of processors, but when Intel released their own x86 family processors with 64-bit instructions, most people began to refer to it as x86–64. Shortly after, Intel launched the IA64 processor family, which had a very different architecture and instruction set than the x86–64 processors and never gained popularity (despite being a very nice processor in its own right).