Central Processing Unit (CPU) is the brain of a PLC controller. CPU itself is usually one of the microcontrollers.
Aforetime these were 8-bit microcontrollers such as 8051, and now these are 16- and 32-bit microcontrollers. Unspoken rule is that you’ll find mostly Hitachi and Fujicu microcontrollers in PLC controllers by Japanese makers, Siemens in European controllers, and Motorola microcontrollers in American ones.
CPU also takes care of communication, interconnectedness among other parts of PLC controller, program execution, memory operation, overseeing input and setting up of an output. PLC controllers have complex routines for memory checkup in order to ensure that PLC memory was not damaged (memory checkup is done for safety reasons).
Generally speaking, CPU unit makes a great number of check-ups of the PLC controller itself so eventual errors would be discovered early. You can simply look at any PLC controller and see that there are several indicators in the form of light diodes for error signalization.