The complex plane is a two-dimensional generalization of the real numbers. Complex numbers (a, b) are added just like you'd expect: (a, b) + (c, d) = (a + c, b + d). They multiply a bit differently, though: (a, b)(c, d) = (ac - bd, ad + bc). You can get ordinary real numbers back by setting the second number of the pair to 0.
Usually (a, b) is written $a+bi$, with the understanding that $i$ is not a real number. In fact it acts as a square root of $-1$, which does not exist in the real numbers. You can see this as so: (0, 1)(0, 1) = (0-1, 0+0) = (-1, 0) or $-1+0i.$