Monodromy
The (informal) fact that continuing a function around a loop back to its start can return a different value.
Monodromy, at the informal level of this course, names the phenomenon that continuing a function element analytically along a closed loop -- one that starts and ends at the same point -- does not always return the original value, because different paths around an obstacle (typically a branch point) can accumulate different continuations. This is why a multivalued function such as \log z or \sqrt z cannot be continued to a single-valued analytic function on any domain surrounding the origin: looping once around 0 changes which branch the continuation lands on.
Examples
Continuing \sqrt z analytically once counterclockwise around the origin, starting from \sqrt 1=1, returns the value -1 rather than 1.
Continuing \log z once counterclockwise around the origin adds 2\pi i to the value it started with.