branch point
Some complex functions refuse to settle on a single value: as you walk a small loop around a certain point and return to where you started, the function comes back changed. The square root flips sign, the logarithm jumps by 2 pi i. The point at the centre of that loop, the place whose mere encirclement scrambles the function, is called a branch point. It is the geometric reason a function is multivalued.
Concretely, a branch point is a point such that a function defined and analytic in a neighbourhood around it (but not at it) does not return to its starting value when continued once around a small circle enclosing it. For w = z^{1/2}, circling the origin once multiplies w by e^{i pi} = -1, so z = 0 is a branch point; a second loop restores the value, so this branch point has order two. For w = log z the value never returns no matter how many loops you take, gaining 2 pi i each time — a logarithmic branch point of infinite order. The point at infinity can be a branch point too, found by the same test on the substitution z = 1/t.
Branch points are not removable by cleverness; they are an intrinsic feature of the function, the seed from which its multivaluedness grows. To work with such a function as a single-valued object you must draw a branch cut, a barrier joining branch points that you forbid paths from crossing. Recognising branch points is the first step in any contour integration involving roots, logarithms, or non-integer powers — they dictate where cuts go and which keyhole or dumbbell contour you must use.
For f(z) = sqrt(z^2 - 1) = sqrt((z - 1)(z + 1)), both z = 1 and z = -1 are branch points. Encircling just one of them flips the sign of f; encircling both together leaves f unchanged, because the two sign flips cancel. That cancellation is why a single branch cut drawn between -1 and +1 (a finite segment) suffices to make this function single-valued.
Branch points come in linked sets; how they cancel under encirclement decides where the cuts go.
A branch point is not a pole and is not a removable singularity. At a pole the function blows up but stays single-valued; at a branch point the function may even stay bounded yet fails to return to its value after a loop. Mistaking one for the other derails the whole contour calculation.