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! 

If $f$ has an isolated singularity at $z_0$, then the coefficient of $\dfrac{1}{z-z_0}$ in the Laurent Series is called the residue of $f$ at $z_0$, denoted
$a_{-1}=\operatorname{Res}_f(z_0)$
