What is the difference between stable air and unstable air?

The 'stability of air parcels' refers to whether or not they are prone to
continued rising or sinking, when perturbed slightly. This is an
important concept for the generation of clouds, and in general for
considerations of transport (movement) of air, water vapor and other gases,
vertically.

A parcel of air may be unstable because lifting it slightly may make it
even more buoyant in which case it will continue to rise. Unstable air may
alternatively be able to continue sinking if displaced down slightly. Stable
air behaves oppositely: it falls back if pushed up and bobs up if pushed down.

Whether a parcel of air is stable or unstable has to do with thermodynamics
and such things as the temperature and humidity of the parcel compared to
the surrounding air - so it is a relative property of air parcels. Air may
rise because it is unstable but then reach air where it no longer is
unstable, and stop there.

The theormodynamic properties of air changes as it is exposed to varying
levels of pressure - think of how hot a bicycle pump becomes when you are
using it: You are compressing the air inside and this makes the air hot.
Air that is compressed and allowed to escape - such as air from a steel
compressed air bottle, or air from a tyre - cools. Well, the same thing
happens in Nature - if a parcel of air rises it will expand and thus cool,
and this change of thermodynamic state could now alter its stability.