Each smile curve on a vol surface is typically described by a few parameters. The most important are
1) atmVol aka anchorVol, and
All other curve-parameters are less important. All the curve-parameters are “calibrated” or “tuned” using market quotes.
Skew is a number and basically describes (for a given maturity) the asymmetry of the vol smile. There’s one skew number for each fitted maturity. These numbers are typically negative.
That’s the parametrization along the strike axis. How about along maturity axis? What parameters describe the term structure of vol?
I don’t know for sure, but often a parametric vol surface has a term structure parametrization for each curve-parameter. For example, there’s a term-structure for anchorVol. There’s another term structure for skew. Well, in some of the most sophisticated vol surface models, there’s no such TS parametrization. I guess in practice users didn’t find it useful.