The CPU executes instructions by fetching them from memory, decoding and executing them in a cycle. It has an ALU for arithmetic and logical operations, registers for data, an instruction register, program counter and control unit. There are three main types of instructions - arithmetic/logical, memory transfer, and branch. The fetch-execute cycle involves fetching the instruction from memory into the instruction register based on the program counter, decoding it, executing it by performing the appropriate operation, and updating the program counter.