a branched cover
An ordinary covering map is like a stack of identical sheets over a base — every point downstairs has exactly the same number of points above it, and locally the cover just looks like several disjoint copies. A branched cover is almost this, but allows special points where the sheets are PINCHED together, like the way the squaring map z -> z^2 wraps the plane twice around itself but glues everything at the origin. Branch points are where the perfect sheet-count momentarily collapses.
Precisely, a nonconstant holomorphic map f: X -> Y between compact Riemann surfaces is automatically a branched cover. Away from a finite set of branch points it is a genuine covering of some fixed degree d: every regular value y has exactly d preimages, and d is the degree of f. At special points the local picture is z -> z^e for an integer e = e_p >= 1, the ramification index at the preimage point p; the map wraps e_p sheets together there. A point with e_p > 1 is a ramification point, and its image is a branch point. Over any value the ramification indices of the preimages always sum to d: sum over f(p) = y of e_p = d. To find e_p in practice, write f in local coordinates centered at p and at f(p) and read off the lowest power.
Why it matters: branched covers are how you BUILD and COMPARE Riemann surfaces. The hyperelliptic curve y^2 = (polynomial in x) is a 2-sheeted branched cover of the sphere, branched exactly where the polynomial vanishes; every meromorphic function realizes its surface as a branched cover of the sphere; and the Riemann-Hurwitz formula reads off how the genus jumps from the ramification. An honest point: ramification is a complex-analytic phenomenon with no real analogue this clean — the local model z -> z^e is exactly e-to-1 except at the single pinch point, and orientation makes the wrapping unambiguous.
The map f(z) = z^2 from the sphere to the sphere is a degree-2 branched cover. Over a generic value w there are two preimages (the two square roots), but over w = 0 and w = infinity the two sheets pinch to one point each (e = 2). So there are exactly two ramification points, at 0 and infinity, and everywhere else the cover is honestly 2-to-1.
z -> z^2 is a 2-sheeted branched cover with ramification at 0 and infinity, where the two sheets pinch.
Don't confuse a branch point (downstairs, in the target) with a ramification point (upstairs, in the source) — many texts use the words loosely. Also, a branched cover is NOT a covering map in the topological sense: the covering-space theory only applies after deleting the branch points.