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.

A function $f$ analytic in a full disk $D_r$ around $a$ can be written as derivative of some other function $F$ in $D$:$$f(z)=F'(z)\quad(z\in D_r)\ .$$
If $f$ has an isolated singularity at $a$ one may still ask whether $f$ has a primitive $F$ in the punctured disk $\dot D_r:=D_r\setminus\{a\}$, in other words: whether the ODE $y'=f(z)$ has a  solution in $\dot D_r$. It turns out that the sole obstruction to the solvability of this problem is the residue
$${\rm res\,}_a(f):={1\over2\pi i}\int_{\partial D_\rho} f(z)\>dz,\qquad\rho<r\ .$$
If this residue is $\ne0$ no solution exists.
In my history of math book (by Moritz Kline) I read that the name of "residue" has been introduced by Cauchy in his Exercices de mathématique (1826–30) in connection with an integral similar to the above, but around a rectangle.
