The physical manifestation of leaving group ability is the rate at which a reaction takes place. Good leaving groups give fast reactions. By transition state theory, this implies that reactions involving good leaving groups have low activation barriers leading to relatively stable transition states.
It is helpful to consider the concept of leaving group ability in the case of the first step of an S1/E1 reaction with an anionic leaving group (ionization), while keeping in mind that this concept can be generalized to all reactions that involve leaving groups. Because the leaving group bears a larger negative charge in the transition state (and products) than in the starting material, a good leaving group must be able to stabilize this negative charge, i.e. form stable anions. A good measure of anion stability is the pK of an anions conjugate acid (pK), and leaving group ability indeed generally follows this trend, with a lower pK' correlating well with better leaving group ability.
The correlation between pK and leaving group ability, however, is not perfect. Leaving group ability represents the difference in energy between starting materials and a transition state (ΔG) and differences in leaving group ability are reflected in changes in this quantity (ΔΔG). The pK, however, represents the difference in energy between starting materials and products (ΔG°) with differences in acidity reflected in changes in this quantity (ΔΔG°).  The ability to correlate these energy differences is justified by the Hammond postulate and the Bell–Evans–Polanyi principle.  Also, the starting materials in these cases are different. In the case of the acid dissociation constant, the "leaving group" is bound to a proton in the starting material, while in the case of leaving group ability, the leaving group is bound to (usually) carbon. It is with these important caveats in mind that one must consider pK to be reflective of leaving group ability.  Nevertheless, one can generally examine acid dissociation constants to qualitatively predict or rationalize rate or reactivity trends relating to variation of the leaving group.  Consistent with this picture, strong bases such as  and  tend to make poor leaving groups, due their inability to stabilize a negative charge.
What constitutes a reasonable leaving group is dependent on context.  For S2 reactions, typical synthetically useful leaving groups include , , and .  Substrates containing phosphate and carboxylate leaving groups are more likely to react by competitive addition-elimination, while sulfonium and ammonium salts generally form ylides or undergo E2 elimination when possible.  With reference to the table above, phenoxides () constitute the lower limit for what is feasible as S2 leaving groups: very strong nucleophiles like  or  have been used to demethylate anisole derivatives through S2 displacement at the methyl group.  Hydroxide, alkoxides, amides, hydride, and alkyl anions do not serve as leaving groups in S2 reactions.
On the other hand, when anionic or dianionic tetrahedral intermediates collapse, the high electron density of the neighboring heteroatom facilitates the expulsion of a leaving group.  Thus, in the case of ester and amide hydrolysis under basic conditions, alkoxides and amides are commonly proposed as leaving groups.  For the same reason, E1cb reactions involving hydroxide as a leaving group are not uncommon (e.g., in the aldol condensation).  It is exceedingly rare for groups such as  (hydrides),  (alkyl anions, R = alkyl or H), or  (aryl anions, Ar = aryl) to depart with a pair of electrons because of the high energy of these species.  The Chichibabin reaction provides an example of hydride as a leaving group, while the Wolff-Kishner reaction and Haller-Bauer reaction feature unstabilized carbanion leaving groups.