[mabcat]

Stability is how the aircraft reacts to disturbances (control inputs, turbulence, ...). Stable aircraft resist disturbances and settle into going where they're pointed at a predictable speed, so you can let go of the controls and attend to other pilot things. Unstable aircraft, when disturbed, go more and more in the direction of the disturbance. (e.g. a turn gets tighter and tighter, nose oscillates up and down more and more, ...) You need to correct this with more control inputs, but these will also have unstable results. So they only let computers fly unstable aircraft.

A car has good stability in turns: it wants to go straight, if there's a bump or you jerk the wheel a bit it still goes straight. If you apply a control input and get a turn, it will let you maintain the turn but it also wants to unwind and go straight. This is stability but not aerodynamic stability.

The reason one might want a less-stable aircraft is that stability is the opposite of manoeuverability. Super-stable aircraft like training gliders really resist turning and can feel too much like hard work.

As another commenter mentioned the wings don't contribute much to stability. The two big fins at the back (the stabilizers) are the main source.

(disclaimer: I fly fixed-wing acft and gliders, barely, and am no physicist.)