Stability is the tendency of an airplane, after a disturbance such as a wind gust, to return to its original flight attitude without control inputs by the pilot.  

Positive Static Stability means that aircraft has an initial tendency to return to its trimmed condition (like a marble in a saucer).

Neutral Static Stability means that an aircraft will tend to stay in its new condition (like a marble on a flat surface).

Negative Static Stability means that an aircraft is unstable and will diverge further when disturbed (like a marble on a convex dish)

Positive Dynamic Stability means that an aircraft that is disturbed will experience an oscillation that is quickly damped.

Neutral Dynamic Stability means that an aircraft that is disturbed will follow a continuing, undamped oscillation.

Negative Dynamic Stability means that an aircraft that is disturbed will have increasingly divergent oscillation.

Longitudinal Stability is in the pitching plane and occurs about the lateral axis.

Directional Stability is an airplane's ability to weathervane its nose into any airflow from the side.

Lateral Stability is the natural ability of the airplane to recover from a disturbance in the lateral plane (rolling about the longitudinal axis). Dihedral wing construction (in which the wings are angled upward from their meeting point in the center of the fuselage) improves lateral stability. If one wing dips down it will meet the relative airflow at a larger angle of attack and generate more lift. The upper wing will meet the relative airflow at a smaller angle of attack and will produce less lift. Thus, the aircraft will have a tendency to return to a wings-level position.

Training aircraft are designed to be statically and dynamically stable, which makes it easier to recover from an unusual attitude. The tradeoff for this built-in safety feature is maneuverability. For this reason, advanced fighter planes are often designed to be less stable but provide greater maneuvering capabilities to the skilled pilot.