What are registers, and how are RAX, EAX, AX, and AL related in x86-64?

Registers hold values used directly by instructions. In x86-64, RAX names a 64-bit general-purpose register, EAX its low 32 bits, AX its low 16 bits, and AL its low 8 bits. These names refer to overlapping storage, not four independent variables. Writing EAX clears the upper 32 bits of RAX, while ordinary writes to AX or AL leave higher bits unchanged. Instruction availability and operand sizes depend on execution mode. These questions use Intel syntax and explicitly identify 64-bit examples.