The leveling effect is a term in physical chemistry to refer to the tendency of any acid or base stronger than a certain threshold to dissociate in a given solvent. All solvents are only capable of handling a maximum strength acid or base. Anything stronger will autoionize to yield something equal to that strongest capable acid/base, or weaker.

As an example, the strongest acid that can exist in water is the hydronium ion, H3O+. Any acid stronger than hydronium will make every effort possible to become hydronium, dissociating. So when one adds hydrochloric acid to water, for example, the reaction goes as follows.

HCl + H2O --> H30+ + Cl-
Hydrochloric acid is much stronger than water, so the leveling effect kicks in and reduces it to hydronium. Likewise for bases, the strongest of which that can exist in water is hydroxide. Throw sodium hydroxide in there aaaand...
NaOH + H2O --> Na+ + OH-
The leveling effect reduces sodium hydroxide to hydroxide and a sodium ion. Although this works with any solvents (and others have much higher thresholds than water), it is most often associated with the principle of aqueous acids and bases. Because of the leveling effect, you don't really have hydrochloric acid or sodium hydroxide floating around in water, as would be predicted by Arrhenius Theory. Instead, you have lots and lots of the threshold acid or base and extra ions capable of forming hydrochloric acid or sodium hydroxide once the water (or any solvent) is removed, as predicted by Brønsted-Lowry Theory.