the image of a region or curve
When a function moves the plane, you usually do not care about a single point — you care about where a whole shape ends up. The image of a set S under f, written f(S), is simply the collection of all outputs f(z) as z runs over S. If S is a curve, its image is the new curve traced out; if S is a region (a filled-in area), its image is the new region it gets sent to.
Finding an image is the bread and butter of the mapping viewpoint, and there are two reliable tricks. The cleanest is to parametrize the shape and push the parameter through f. To find the image of the unit circle under f(z) = z^2, write z = e^(i theta) for theta from 0 to 2 pi; then w = z^2 = e^(2 i theta). As theta goes once around, 2 theta goes twice around, so the image is the unit circle traced twice. The other trick is to translate a condition on z into a condition on w: under w = 1/z, the line Re(z) = 1 becomes a circle through the origin in the w-plane, found by substituting z = 1/w and simplifying.
Images are where the deep theorems live. The open mapping theorem says a non-constant holomorphic function sends open regions to open regions; conformal maps preserve the angles at which curves in an image cross; and the whole programme of solving a hard boundary-value problem by mapping its region to a simpler one (a disk or half-plane) is image-chasing in disguise. Picture the image first, then compute.
The horizontal strip 0 < Im(z) < pi maps under f(z) = e^z onto the entire upper half-plane Im(w) > 0: the bottom edge y = 0 goes to the positive real axis, the top edge y = pi to the negative real axis, and the strip between fills everything above.
Images of strips, lines and circles are the everyday currency of conformal mapping.
The image of an open set under a continuous map need not be open in general — that strong property is special to non-constant holomorphic functions (the open mapping theorem). Continuity alone only guarantees the reverse: pre-images of open sets are open.