A half adder is a logic circuit that performs one-digit addition. It requires two inputs, and the output is the sum of the inputs. An example of a Boolean half adder is this circuit:
So, as you can see, the output of the whole circuit is the sum digit (the ones digit), and the output of the AND gate is the carry digit (the twos digit). Thus, according to the half-adder, 1 + 1 = 10 (keep in mind that this is in base 2). Likewise, 0 + 1 and 1 + 0 both equal 01, or 1.
If you want to add two multiple-digit numbers, you must use a full adder. A full adder is basically two (or more) half adders linked together. For the purposes of this page you won't really need to know about full adders, but if you would like to know more about them this page has a good (albeit advanced) explanation of full adders.