A Variant Class is a mechanic that is applied in RPGs with a distinct class-structure approach; more freeform games, such as Unknown Armies, GURPS, Deadlands, World of Darkness, etc, would have no use for it.
As the name suggests, a variant class is an alternative set of traits and features applied to an existing class for a different interpretation of the class's core concept whilst still retaining greater cohesion. Variant classes are simpler and easier to use than full-fledged alternate classes, because they "piggyback" most of their traits on the existing class. Dungeons & Dragons has long made extensive use of this practice, from the Kits of Advanced Dungeons & Dragons to the openly named variant classes and (closely related but not quite the same) alternate class features of 3rd Edition to the "Archetypes" mechanics of Pathfinder and 5th Edition.
Examples of variant classes in D&D include the Sidhe Scholar (a Druid who learns its lore from fae tutors, giving it unique traits like learning Evil Domain spells) and the Paladins of Freedom, Tyranny and Slaughter (Paladins who must be Chaotic Good, Lawful Evil and Chaotic Evil, respectively). Similarly, 4th Edition has the oft-maligned Essentials classes from the latter parts of the edition's lifespan (Usually from the Heroes of X classes) which attempted to simplify the progressions of classes by making overarching features that progressed as the character advanced, but often stripped the character of any choice from the parent class' powers and sticking them with leftovers.