Vorticity is a vector measure of local rotation in a fluid, defined mathematically as the curl of the velocity field. In meteorology, the word usually refers to the vertical component of that vector, which describes rotation about a vertical axis at a point in the atmosphere.
What the equation is saying
The formal definition is that vorticity equals del cross the velocity vector. For solid rotation, this works out to exactly twice the angular velocity. Physically, if you dropped a tiny paddle wheel into the flow with its axis vertical, the vertical vorticity is proportional to how fast it would spin.
Vorticity arises in two ways in a horizontal flow. Curvature vorticity comes from the flow curving through a bend, like the base of a trough. Shear vorticity comes from a change in wind speed across the flow, like faster winds on one side of a jet streak than the other. Both count toward the same total.
Relative and absolute forms
Relative vorticity is what you compute from the winds observed on a rotating Earth, without adding any correction. Absolute vorticity is relative vorticity plus the Coriolis parameter, which represents the background spin of the Earth itself at that latitude. Absolute vorticity is what is conserved for large-scale, barotropic flows and is usually what the 500 mb vorticity chart is showing.
The Coriolis contribution grows with latitude and is zero at the equator. In the Plains and Southwest, absolute vorticity values on a 500 mb chart include a nontrivial baseline from Earth's rotation before any weather system contributes.
Where forecasters see it
On synoptic scales, 500 mb vorticity maxes mark the shortwaves that drive broad ascent and precipitation. Downstream of a vorticity max, differential positive vorticity advection through a layer forces vertical motion under the quasigeostrophic framework, which is the classical mechanism for lift under an approaching upper disturbance.
On storm scales, vertical vorticity is the language of rotation. Mesocyclones are identified by strong, persistent positive vertical vorticity in a supercell updraft. Near-ground vertical vorticity underneath a rotating storm is a necessary ingredient for tornadogenesis, though it is not sufficient by itself.
Common misreads
A bull's-eye of vorticity on a model chart is not automatically a storm producer. It represents a rotation of the flow field at that point, which may or may not project onto ascent, moisture, and instability that produce weather. The vorticity is one ingredient in the vertical-motion equation, not the whole story.
At storm scale, a large radar-derived rotational value in one volume scan does not necessarily indicate a persistent mesocyclone. Vorticity has to be sustained through multiple volumes and connected to a coherent updraft before it says something about tornado potential.