Since two's complement has been so nicely explained by SoberSephiroth above, this is now a good spot to point out the following bit of fun involving infinite series:

Everybody knows that the infinite sum:

S = 1 + 1/2 + 1/4 + 1/8 + ...

is equal to 2. One way to see this immediately is to double it, so:

2S = 2 + 1 + 1/2 + 1/4 + ...

and then subtract. All the other terms cancel, leaving S = 2. But you have to be careful when you do this sort of thing to make sure the series you're manipulating all converge. Let's try that again with:

T = 1 + 2 + 4 + 8 + ...

now we double it again:

2T = 2 + 4 + 8 + 16 + ...

and subtracting we get T = -1, even though we know T is unbounded. Yikes!

There's a wonderful margin note in the book Concrete Mathematics by Graham, Knuth, and Patashnik (yes, that Knuth) where they talk about the sums above. Next to the yikes! result, it says: "Sure: 1 + 2 + 4 + 8 + ... is the "infinite precision" representation of the number -1, in a binary computer with infinite word size".