Register banking refers to providing multiple copies of a register at the same address.
Taken from section 1.4.6 of the arm docs
The term is referring to a solution for the problem that not all registers can be seen at once.
There is a different register bank for each processor mode. The banked registers give rapid context switching for dealing with processor exceptions and privileged operations.
If your looking for a more theoretical reasoning, I recommend this paper.
Edit: A much deeper answer than mine is given here