Difference between x86, x64 and ARM architecture

Both x86 and x64 are same CPU architecture. HOW?

x86 is a "CPU" architecture which was initially used with 16bit "chip" and later on extended to be used on 32 bit chip. For most of the time, we used a 32bit chip(each number represented with 32 bits) that had x86 architecture. So basically, when we said this computer (CPU) is x86, it meant 32 bit or vice versa and same with OS, if an OS was 32bit, then it meant it was running x86 CPU architecture chip. (I'm not considering ARM here)

Now, due to the limitation of memory usage with 32bit chip that it can only support 4GB of RAM, x86 was then extended to be used on 64 bit chip, so basically now the same architecture (instructions set) was used to build a chip that used 64bit to represent a number, hence 64bit chip. Initially it was called x86-64 and later reduced to x64 which meant that x86 is the architecture on a 64bit chip.

Like x86 and x64, ARM is a different processor (CPU) architecture. The ARM architecture is typically used to build CPUs for a mobile device, ARM64 is simply an extension or evolution of the ARM architecture that supports 64-bit processing. Devices built on the ARM64 architecture include desktop PCs, mobile devices, and some IoT Core devices (Rasperry Pi 2, Raspberry Pi 3, and DragonBoard). For example the Microsoft HoloLens 2 uses an ARM64 processor.

About the architectures, you can refer to this document.