Take an initial guess at where f(x) = 0; call it x1. Then you generate what you hope is a better guess x2 by finding where the line through (x1, f(x1)) with slope f'(x1) hits zero. Lather, rinse, repeat.

Things can get ugly--say, if the derivative is zero at your guess--and if your initial guess is bad, or f is wonky (say it has several zeroes near one another) it may take a long time. But as people have said, if you have a good guess, convergence is quadratic, doubling the number of good bits each time, so for floating point square root, peek at the exponent and take half that (multiply by sqrt(2) if it's odd). log2(#mantissa bits) iterations and you're as good as it gets.

Apparently this new method avoids the nasty cases and is still fast. Cool stuff.