Pokémon-EX are Pokémon that have been blessed by a legendary with higher stamina and some number of unique moves. They also happen to look noticeably different from others of their species.
This is a concept brought from the Trading Card Game, where the features of special Pokémon cards of various generations (Pokémon-X and Pokémon LV.X, among others) have been mixed and adapted into the format of the RP.
A Pokémon becomes a "Pokémon-EX" by blessing from a Legendary, signifying this Pokémon is trusted to take care of certain affairs on behalf of the Legendary, or that the Legendary acknowledges the Pokémon's contribution in an important event. The Pokémon in question is usually called before the presence of a Legendary to be bestowed this title. This is different from variations such as shininess (which is accidental) or being a Shadow Pokémon (which is imposed onto the individual by force).
The transformation basically grants these Pokémon a somewhat higher stamina, recognizable markings, and a unique move. Beyond the markings, the Pokémon remains the same physiologically speaking - they can still breed normally and are still recognized by their Pokéball. Their Aura is barely altered.
As a nod to the TCG rule of double prize for defeating a EX card, when a Pokémon-EX is defeated, it grants double experience.