Binary addition is the column addition you learned at school, with a smaller base. Start at the right, add each column, and carry into the next column when the total doesn't fit in one digit.
A binary digit only holds 0 or 1, so any column total of 2 or more carries. Including the carry from the right, a column adds at most three bits:
- 0 + 0 = 0
- 0 + 1 = 1
- 1 + 1 = 2 =
10₂: write 0, carry 1 - 1 + 1 + 1 = 3 =
11₂: write 1, carry 1
Write each carry above the next column so you don't lose it.
In hardware, this one-column job is a full adder, and a chain of them is a ripple carry adder. When the numbers have a fixed width, a carry out of the top column has nowhere to go: that's the carry out. For unsigned numbers it means the sum didn't fit (unsigned overflow).
The same procedure adds twos complement numbers with no changes. Only the overflow check differs.