The Wikipedia article you have sited states a simple but clear interpretation of this quantity and some basic use:
"The enstrophy can be interpreted as another type of potential density (ie. see probability density); or, more concretely, the quantity directly related to the kinetic energy in the flow model that corresponds to dissipation effects in the fluid. It is particularly useful in the study of turbulent flows, and is often identified in the study of thrusters as well as the field of flame theory."
This seems to answer your basic queries. However, I understand from the previous answer that you want a bit more information on this quantity and its use. Well...
I don't believe I can define enstrophy any more clearly that you have yourself (and Wikipedia does) already, however, let’s try, I can then give you some idea of its use. Local anisotropic behaviour of enstrophy can be treated as the self-organization of the regions of high vorticity in coherent vortex structures – most notably vortex filaments/tubes. This turbulent behaviour is ubiquitous and still not very well understood. In particular, local alignment or anti-alignment of the vorticity direction, i.e. local coherence, is prominently featured in turbulent flows.
It is thought, but still an area of active research, that evolution of voticity in turbulent flows leads/is governed by 3D enstrophy cascades on 'large-scales' of the turbulence. I am no expert in this area but have read a few papers on this when writing a 2D code for turbulence modelling.
I hope this helps.