I keep running across references to 2's complements, does anyone have a
    relatively simple explanation?

Twos complement is one way to represent signed numbers in a digital system
in such a way that the same hardware logic can do arithmetic without having
to understand the sign-ed-ness.

To get the twos complement of a number, invert all the bits and add 1.
(most procesors will have a "negate" instruction to do this.)

Say you have +1 and -1.  If you're a human, and you want to add them
together, you note that one is negative and do a subtract instead.  However,
that's a bit much to ask of combinatorial hardware logic.

So, instead, we let +1 be 00000001, and -1 = (11111110 + 1) = 11111111.
If you add these together, you'd get 100000000, or simply 00000000 if
you neglect the carry bit.  Hey - exactly what you wanted...

An alternative technology was "ones complement", where you simply invert
all the bits.  In this case, to make your math come out right, you add
the carry bit back into the result.  So 00000001 + 11111110 (-1) =
11111111, which is "-0" (oops - that's one reason ones complement died
out.  Having two values of zero is confusing.)

BillW