a multiply connected domain
If a simply connected domain is a region with no holes, a multiply connected domain is one with holes — a region that is connected but where some loops cannot be shrunk to a point because a hole gets in the way. The simplest example is an annulus, a flat ring between two concentric circles: a loop going once around the central hole is trapped. The number and arrangement of holes is exactly what 'multiply connected' is tracking.
Holes change everything for integration. On a multiply connected domain, Cauchy's theorem in its simple form no longer guarantees that loop integrals vanish, because a loop may encircle a hole that hides a singularity of f. But a precise replacement holds: if f is holomorphic on the domain, the integral around the outer boundary equals the sum of the integrals around the inner boundaries (each oriented consistently), since the difference can be cut into pieces lying in a hole-free part. This is the 'deformation' or 'multiply connected Cauchy theorem', and it is the doorway to residues — each hole contributes its own bookkeeping.
The honest content is that the value of a loop integral now depends on which holes the loop wraps around and how many times. A loop circling a single puncture once may give 2 pi i times a residue; circling it twice doubles that; circling a different puncture gives a different number. The plane minus several points is the standard multiply connected playground, and the residue theorem is precisely the accounting system for summing the contribution of every hole a contour catches.
On the twice-punctured plane (with 0 and 1 removed), f(z) = 1/(z(z-1)) is holomorphic. A loop enclosing only 0 gives one value; a loop enclosing only 1 gives another; a large loop enclosing both gives the sum of the two — the two holes contribute additively, which is the deformation theorem in action.
On a multiply connected domain a loop integral splits into a sum over the holes it encloses.
Connected and simply connected are different: a multiply connected domain is still in one piece — what fails is that not every loop can shrink to a point.