Arithmetic logic unit (ALU)
The arithmetic logic unit (ALU) part of a CPU is usually capable of performing various arithmetic, bitwise, and comparison operations on binary numbers. In our simple breadboard CPU, the ALU is just able to add and subtract. It's connected to the A and B registers and outputs either the sum of A+B or the difference of A-B.
Theory
Before diving into building the ALU, check out these two videos. The first shows how we can use hardware to add numbers in binary. The second video explains how negative numbers work. Between the two videos, you'll know how to add negative numbers, which means you'll know how to subtract!
ALU design and construction
Schematic

The flags register portion of the schematic is described later on as part of the CPU control logic.
Data sheets
ALU:
Flags register: