The advection equation is the partial differential equation that governs the motion of a conserved scalar field as it is advected by a known velocity vector field.  It is derived using the scalar fields conservation law, together with Gausss theorem, and taking the infinitesimal limit.
One easily  visualized example of advection is the transport of ink dumped into a river. As the river flows, ink will move downstream in a "pulse" via advection, as the water's movement itself transports the ink. If added to a lake without significant bulk water flow, the ink would simply disperse outwards from its source in a diffusive manner, which is not advection.  Note that as it moves downstream, the "pulse" of ink will also spread via diffusion.  The sum of these processes is called convection.