a parametrized path

Before you can integrate along a curve you need an honest description of the curve itself. A parametrized path is a curve in the plane given by a continuous function z(t) of a real parameter t running over an interval, say from a to b. As t increases, the point z(t) traces out the curve, and the direction of increasing t fixes an orientation — a sense in which you travel along it. Think of t as a clock and z(t) as where a walker stands at time t.

For contour integration we want the path to be reasonably tame: a 'smooth' path has a continuous, nonzero derivative z'(t), so it has a well-defined tangent direction everywhere. A 'contour' (or piecewise-smooth path) is a finite chain of smooth pieces joined end to end — for instance three straight segments forming a triangle, or four forming a rectangle. We allow corners, but not wild behavior. A path is 'closed' if z(a) = z(b), meaning it returns to its start, and 'simple' if it does not cross itself in between.

The same geometric curve can be parametrized in many ways — fast or slow, with different clocks — and reparametrizing in an orientation-preserving way does not change a contour integral. What does matter is the curve as a set together with its orientation. This is why people speak loosely of 'the contour C' while quietly carrying a parametrization in their pocket to do the actual computation.