sheets and branch points
When you build a Riemann surface by stacking copies of the plane, each copy is a sheet — a single layer carrying one branch of the multivalued function. Over most points of the plane the surface has several sheets sitting one above another, and on each sheet the function takes one of its several values. A branch point is a special location where the sheets do not just stack but actually cycle into one another: as you make a small loop around that point, you slide from one sheet onto the next, and only after enough loops do you return to where you began. Branch points are the rivets that fasten the sheets into one connected surface.
The local picture at a branch point is precise. Near a branch point of order m (where m sheets cycle), the surface looks like the map w to w^m: think of m sheets joined so that going once around the base point advances you one sheet, and m circuits bring you back. For sqrt(z) the point z = 0 is a branch point of order 2 — two sheets, one loop flips you to the other sheet, a second loop restores you. For the cube root z = 0 has order 3. For the logarithm z = 0 is a branch point of INFINITE order: each loop carries you to a brand-new sheet (adding 2 pi i), so the sheets form an endless spiral that never closes up. The point at infinity can also be a branch point, and you check it by examining the function in the coordinate 1/z.
Reading off sheets and branch points is how you understand the global shape of a multivalued function. The NUMBER of sheets tells you how many values the function has (two for square root, n for an n-th root, infinitely many for log). The branch points and the way sheets cycle around each one encode the monodromy — exactly which branch you land on after each loop. Knowing this combinatorial data (sheets plus the branching behaviour at each branch point) is enough to reconstruct the whole Riemann surface, and from it the genus and topology of the surface.
For w = sqrt((z-1)(z+1)), the branch points are z = 1 and z = -1 (each of order 2), and there are two sheets. Make a loop around just z = 1: you switch sheets, so sqrt flips sign. But a loop enclosing BOTH branch points flips you twice — back to the original sheet. Gluing the two sheets along a cut joining -1 to 1 produces a surface that turns out to be a sphere.
Branch points are where sheets cycle; a loop enclosing two order-2 points returns you to the start. This surface is a sphere.
A branch point is genuinely different from an ordinary singularity like a pole: around a pole the function is single-valued (it just blows up), but around a branch point the function changes value, returning to itself only after several loops. The number of sheets equals the number of values, which for the logarithm is infinite.