What is 1+1?

We have two inputs, A and B. Each one is 0 or 1. Let's just work through every possible combination.
0+0 = 0. Fine.
0+1 = 1. Fine.
1+0 = 1. Fine.
1+1 = ?

In binary, there is no 2. The only digits are 0 and 1. So 1+1 can't be written as a single digit. Instead it becomes 10. A 0 in this column, and a 1 that moves over to the next column to the left.

You've done this before, just in decimal. When you add 9+1 you don't write 10 in the same column. You write 0 and carry the 1. Same idea here, just with smaller numbers.
Every addition has two parts: what stays in this column, and what carries over.

We call them the sum and the carry.