Intuition about residues of holomorphic functions

Question

Does the concept of residue go beyond the identification with the coefficient of the term $z^{-1}$ of the Laurent series of a holomorphic function? Wouldn't the name for this algebraic entity give a clue? Is there any geometric/physical idea to understand this number? I imagine a residue as a obstruction of the domain for the apply the cauchy-goursat theorem; for example, for every closed curve, the integral of holomorphic function is zero; it means for me which is possible to deform the loop in a point and thus the output of integral is zero.