A register is a very small amount of very fast memory that is built into the CPU (central processing unit) in order to speed up its operations by providing quick access to commonly used values.
Memory refers to semiconductor devices whose contents can be accessed (i.e., read and written to) at extremely high speeds but which are held there only temporarily (i.e., while in use or only as long as the power supply remains on). Most memory consists of main memory, which is comprised of RAM (random access memory) chips that are connected to the CPU by a bus (i.e., a set of dedicated wires).