Definition. See also here. It is a Smooth function between a Differentiable manifold and its Tangent bundle. – Definition on a general manifold – requires defining the Tangent bundle
By looking at them as the Differential of a One-parameter group of diffeomorphisms one can see vector fields as representing an infinitesimal flow. 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.
Vector fields as derivations. Vector fields can be seen as exactly those Linear maps which satisfy the Leibnitz rule.