Can anyone explain a residue in fairly simple terms? I'm studying Complex Analysis and everything up to this point has been pretty straightforward to visualise, but I can't get my head around residues, especially as they seem to have two very different definitions (as a Laurent series coefficient and as an expression involving an integral on a closed path) that I can't understand why they equate. Can anyone give a fairly simple explanation of residues?
Thanks! 

To see why they equate, just take the following contour integral around the unit circle:
$$
\oint \frac{dz}{z^k}=\int_{0}^{2\pi}\frac{d(e^{i\theta})}{e^{ik\theta}}=\int_{0}^{2\pi}\frac{ie^{i\theta}d\theta}{e^{ik\theta}}=\int_{0}^{2\pi}ie^{i(1-k)\theta}d\theta.
$$
If $k=1$, then the integral is $2\pi i$.  If it's any other integer, then it's
$$
\frac{ie^{i(1-k)\theta}}{i(1-k)}\bigg\vert_{0}^{2\pi}=0,
$$
because the integral is periodic with period $2\pi$.  Only the $\frac{1}{z}$ term contributes to a contour integral around a point (with a factor of $2\pi i$); and it gets a special name for being so special.
