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.
Instructions include data-handling instructions (such as arithmetic and logical operations), and control instructions (such as jumps, both conditional and un-conditional).
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.