Indented contour
A contour with a small detour (usually a tiny semicircle) routed around a singularity that sits on it.
An indented contour is a contour that has been modified with a small detour -- typically a tiny semicircular arc -- to route around a singularity that would otherwise lie directly on the path, most often a simple pole sitting on the real axis. Letting the indentation's radius shrink to zero turns the detour's contribution into a computable limit (a fraction of 2\pi i times the residue there, for a simple pole), which is exactly how real improper integrals with a singularity on the line of integration get evaluated.
Examples
To integrate \dfrac{\sin x}{x} over the real line, the contour is indented with a small semicircle above (or below) the pole of 1/z at x=0.
As an indentation's radius \varepsilon\to 0 around a simple pole z_0, the integral over that small semicircular arc approaches \pm i\pi\operatorname{Res}_{z=z_0}f, with the sign set by the direction of the detour.