Consider the second digit of the binary number. First the inputs A and B are added. Furthermore the transfered information from the first digit, the Carry Bit, has to be added. Hence the circuit gets a third input, the CarryIN CI. Such a logical unit is called a full adder.

There are many ways to build a full adder using logic gates. The circuit used here consists basically of two half adders. As a result the logic can be traced through the circuit more easily.
Now back to the problem of two 4bit numbers to be added.