a simply connected domain
Picture a region drawn on a page. It is simply connected if it is connected (in one piece) and, intuitively, has no holes — any loop you draw inside it can be shrunk continuously down to a point without ever leaving the region. A filled disk is simply connected; a disk with a dot punched out of its center is not, because a loop around the missing dot is snagged and cannot contract past it.
More carefully: a domain (a connected open set) is simply connected when every closed curve in it is null-homotopic — it can be deformed continuously, staying in the domain the whole time, to a constant point. Equivalent everyday pictures: the region has no holes, no islands removed, no slits that a loop could wrap around. The whole plane, a half-plane, the interior of any disk or rectangle, and any convex or star-shaped region are all simply connected. The plane with a point removed, an annulus (a ring), and the plane with a line segment deleted are not.
This topological notion is the exact hypothesis Cauchy's theorem needs. On a simply connected domain, a holomorphic function's loop integrals all vanish, it has a primitive, and its integrals are path-independent — because there is no hole for a loop to trap a singularity around, and any loop can be shrunk to a point (where its integral is trivially zero) through a region where f stays holomorphic. Remove that hypothesis and the theorem genuinely breaks, as the punctured plane and 1/z show.
The open right half-plane (all z with positive real part) is simply connected, so any function holomorphic there — including a branch of log z, since 0 is excluded — has loops that vanish and a primitive. The annulus 1 < |z| < 2 is not simply connected: a loop going once around the inner hole cannot be contracted, and indeed the integral of 1/z around it is 2 pi i.
Simply connected (no hole, loops contract) versus an annulus (a hole snags loops).
Simple connectivity is purely about the shape of the domain, not about any function on it — yet it is exactly the shape condition that makes every holomorphic function on the domain behave perfectly under integration.