Vorticity is the localized rotation of the air. Air that rotates counterclockwise, such as in cyclones and troughs, is said to have positive vorticity. Clockwise rotating air, such as in high pressure systems and ridges, has negative vorticity. The advection of vorticity at high levels will result in a response at the surface which will attempt to offset the effects of the advection. More specifically, vorticity advection is indicative of rising motion/falling pressures at the surface. For example, look at this 500 mb map for 12Z, October 29, 1995.
Now look at these two maps of surface pressure (solid lines) from 12Z October 29,1995 and 0Z October 30,1995.
Notice how the surface low has deepened in the area of strong vorticity advection.