a prime end
When you conformally map a disk onto a region with a complicated boundary — one that pinches, spirals, or touches itself — a puzzle appears at the edge. The boundary circle of the disk has a perfectly clean set of points, but the target boundary may be tangled, so 'which boundary point does this disk-edge point go to?' has no good answer in terms of ordinary points. Prime ends are the repair: they are the right notion of 'boundary point' that makes the boundary correspondence work no matter how wild the region.
Caratheodory introduced prime ends to make sense of boundary behaviour of conformal maps. The idea: instead of ordinary boundary points, use equivalence classes of shrinking chains of crosscuts — nested arcs that cut deeper and deeper toward a piece of the boundary, sorted so that two chains are equivalent when they squeeze down to the same boundary approach. Each such class is a prime end. The payoff theorem: a conformal map of the disk onto any bounded simply connected region extends to a homeomorphism between the closed disk's boundary (ordinary points) and the space of prime ends of the region. So the abstract boundary always matches the circle one-to-one, even when the literal boundary does not.
Why bother: an ordinary boundary point of a nasty region can be approached along genuinely different routes that 'should' be different boundary points (think of a slit, where the two sides of the cut are distinct prime ends even though they are the same point of the plane). Prime ends separate these. The caution: a single geometric boundary point can correspond to SEVERAL prime ends (the two banks of a slit, or each side of a pinch), and conversely a prime end can be 'spread out'; the prime-end boundary is generally not the same as the topological boundary. It is the correct, finer object for boundary correspondence.
Map the disk onto the slit disk { |w| < 1 } minus the segment [0, 1). The single geometric point at, say, w = 1/2 on the slit is approached from above and from below by genuinely separate routes, and these are TWO distinct prime ends. The boundary circle of the source disk wraps around the slit, visiting each side as a different prime end.
On a slit, the two banks of the cut are different prime ends though they share one point of the plane.
A prime end is NOT the same as an ordinary boundary point: one point of the plane can be several prime ends (the two sides of a slit). The conformal boundary correspondence is a homeomorphism onto the space of prime ends, not onto the topological boundary.