Contour

A piecewise-smooth curve, traced out in a definite direction, used as the path for a contour integral.

A contour is a piecewise-smooth curve in the complex plane, given together with a parametrization $z(t)$ for $t$ in some interval $[a,b]$ that fixes a direction of travel -- traversed as $t$ increases from $a$ to $b$ . That direction is exactly what a contour integral integrates along, and reversing it negates the integral's value.

Examples

The boundary of a square, traced counterclockwise starting from one corner, is a contour made of four straight-line pieces joined at the corners.
The unit circle $z(t)=e^{it}$ , $0\le t\le 2\pi$ , is a contour with a single smooth piece and no corners at all.