Simply connected
A domain with no holes -- informally, every loop drawn inside it can be shrunk to a point without leaving the domain.
A domain is (informally, at this stage) simply connected if it has no holes: any closed loop drawn inside it can be continuously shrunk down to a single point while staying inside the domain the whole time. A disk is the model example; a punctured disk or an annulus is not, since a loop that circles the missing center or hole can never shrink past it without leaving the domain. Chapter 7 sharpens this into a working test: a domain is simply connected exactly when every simple closed contour drawn inside it encloses only points of the domain itself, with no hole poking through -- precisely the condition Cauchy-Goursat needs to guarantee \oint_C f(z)\,dz=0 for every closed contour in the domain, not merely some of them.
Examples
The disk |z|<1 is simply connected -- every loop inside it can be shrunk to a point.
The punctured disk 0<|z|<1 is not simply connected: a loop circling the origin once cannot be shrunk to a point without passing through the missing center.