the branch obtained around a loop

Take a multivalued function, fix one branch of it near a starting point, and walk that branch once around a closed loop, continuing as you go. When you return to where you started, the function may NOT come back to its original value: you arrive holding a different branch. This is the monodromy of the loop — the rule that records which branch you end up with after a full circuit. It is the most vivid face of multivaluedness: same point, same function, but the looping has shuffled you to another sheet.

The two model examples make it concrete. For the logarithm, start on the principal branch near z = 1 and continue once counterclockwise around the origin. Along the way arg z increases steadily, and by the time you close the loop arg has grown by exactly 2 pi, so log z has increased by 2 pi i. You return to z = 1 not with the value 0 but with 2 pi i — the loop has bumped you up one branch. Go around k times and you pick up 2 pi i k; the branches of log form an infinite ladder indexed by the integers. For the square root, start with one value of sqrt(z) near z = 1 and loop once around 0: arg z grows by 2 pi, so sqrt z = exp((1/2) log z) gets multiplied by exp(i pi) = -1. You return with MINUS your starting value. A second loop multiplies by -1 again and restores the original — sqrt has just two branches that swap each time you circle the branch point.

What governs all this is the loop's homotopy class around the singular points. Two loops that can be deformed into each other (without crossing a branch point) produce the same branch change; a loop that contracts to a point (encircling nothing) returns you to the original branch, by the monodromy theorem. So the assignment 'loop, to branch change' is really a map from the fundamental group of the punctured region into the symmetries of the set of branches — the monodromy representation. For log this group is the integers (add 2 pi i per turn); for the square root it is just two elements (identity and swap).