Lift the phase portrait into three dimensions. At every point in space the field ẋ = f(x) gives a velocity; release thousands of tracer particles and let the field sweep them along. The paths they trace are the flow — spiralling out of a saddle-focus, orbiting a vortex, funnelling into a sink, or winding forever around the two wings of the Lorenz attractor.
A vector field assigns a velocity vector to every point; a flow is what you get when you set particles loose and let that field move them. The three linear flows show the 3-D versions of the phase-portrait fixed points — a saddle-focus spirals outward in a plane while sinking along an axis; a vortex sends everything into circulating orbits; a sink funnels all trajectories into the origin. The Lorenz flow is nonlinear and famous: trajectories never repeat and never cross, yet stay forever on a butterfly-shaped strange attractor — the birthplace of the phrase "butterfly effect." Flows like these are the language of fluid dynamics, weather, and every continuous dynamical system.