What is a residue? I've heard of residues in complex analysis, contour integration, etc. but all I really know it to be is the $c_{-1}$ term in the Laurent series for a function. Is there some sort of intuition on what a "residue" actually is? The terminology makes it seem like something left over, or something like that.

The holomorphic functions have an extraordinary property: if you compute an integral along a path, the value of the integral does not depend on the path !
More precisely, if the function is holomorphic everywhere inside a closed path, the integral is just zero. But if the function has poles (zeroes at the denominator, $c_{-k}$ terms in the Laurent series), every pole brings a nonzero contribution called its residue. You can shrink the path as much as you want, even turning it to infinitesimal circles around every pole, provided you keep the poles in.
So the residues are what is left (as regards integration) after you removed all the holomorphic parts of the domain.

