Central Processing Unit
The Central Processing Unit, usually abbreviated as CPU, or simply called a processor, is the 'brain' of a computer - the unit which controls the operation of the entire computer, by reading instructions from main memory and executing them.
The CPU will contain an ALU, which is where any computation (arthmetical or logical) is performed. The CPU also usually contains a number of registers (storage for data inside the CPU, instead of in the main memory).
One special register, the Program Counter, indicates where the current/next instruction is to be fetched from (although there were early computers which kept this information in a specific location in main memory). CPUs also typically include registers for holding data which is being worked on: early computers often had only one, called the 'accumulator'; and there have been computers without any such data registers.
Finally, CPUs usually contain means for controlling the peripherals attached to the computer; reading data from, and writing data to, them.