the flow of a vector field
A vector field assigns an arrow to every point — a wind blowing across the manifold. Its flow is what happens when you let a particle drift on that wind: at each instant it moves in the direction the field points, tracing out a path. The flow bundles all these trajectories, started from all points at once, into a single time-dependent transformation of the whole manifold.
Given a smooth vector field X on M, an integral curve through p is a curve gamma with gamma(0) = p and gamma'(t) = X(gamma(t)) for all t — it solves the autonomous ODE whose right-hand side is X. The existence-uniqueness theorem for ODEs gives a unique integral curve through each point, at least for a short time. Collecting them defines the flow theta_t(p) = (position at time t of the integral curve through p). For each fixed t, theta_t is a diffeomorphism of (an open subset of) M onto its image, and they compose like time: theta_s composed with theta_t equals theta_{s+t}, with theta_0 the identity — a one-parameter group of diffeomorphisms.
Flows are how vector fields 'integrate' to motions, and they are the geometric meaning behind exponentials and Lie derivatives: the Lie derivative of a tensor along X measures its rate of change as you drag it by theta_t. A crucial caveat is completeness: in general theta_t is only defined for small t, because integral curves can run off to infinity in finite time — the field X(x) = x^2 on R has integral curves that blow up. A vector field is called complete when its flow exists for all t, giving a genuine action of the whole real line; this is automatic on compact manifolds but can fail otherwise, so 'the flow' is often only locally defined.
On R^2 the rotation field X = -y partial-x + x partial-y has flow theta_t(x, y) = (x cos t - y sin t, x sin t + y cos t), rigid rotation by angle t. It is complete: every integral curve is a circle traced forever, and theta_s composed with theta_t = theta_{s+t} is just angle addition.
The rotation field integrates to the rotation flow; it is complete.
Not every vector field is complete: integral curves may escape to infinity in finite time, so the flow can be only local. Completeness is automatic on compact manifolds and on manifolds where X has compact support.