One can find a correspondence between Vector fields (under some assumptions) and a one-parameter group of Diffeomorphisms, so that the vector field corresponds to the "infinitesimal displacement" of the flow represented by this group, the parameter being time.

See here for the formal definition and proof of correspondence. The correspondence uses integral curves to build the One-parameter group of diffeomorphisms given a vector field.

A Curve in the manifold whose Tangent vector at every point is the vector of the vector field at that point is called an Integral curve