Historically, a register was a sign, on which people could write, often with chalk?.
In computers, a memory data register or memory address register are memory locations that can be loaded or stored, and operated upon very quickly by software or a microprogram, using an arithmetic and logic unit of a CPU.
The term is also used in Australia (and, I think, in Britain), as an abbreviation for [cash register]?, a device for tracking retail sales and known to Americans as a till