Epeirogeny ("EPP-ir-rod-geny") is strictly vertical movement of a continent rather than horizontal movement that compresses it to form mountains (orogeny) or stretches it to form rifts (taphrogeny). Instead, epeirogenic movements form gentle arches and structural basins, or they lift whole regions evenly.
In geology school, they don't say much about epeirogeny-it's an afterthought, a catch-all word for processes that aren't mountain-building. Listed under it are things like isostatic movements, which result from the weight of glacial ice caps and their removal, the subsidence of passive plate margins like the Atlantic coasts of the Old and New Worlds, and various other puzzling uplifts that are typically ascribed to mantle plumes.
We'll ignore isostatic movements here because they're trivial examples of loading and unloading (although they account for some dramatic wave-cut platforms). Phenomena related to passive cooling of hot lithosphere also pose no mystery. That leaves examples where we believe that some force must have actively pulled down or pushed up the continental lithosphere (note that it only refers to the continental lithosphere, as you don't see the term in marine geology).
Epeirogenic movements, in this narrower sense, are considered evidence of activity in the underlying mantle, either mantle plumes or consequences of plate-tectonic processes like subduction. Today that topic is often called "dynamic topography," and it could be argued that there is no need for the term epeirogeny anymore.
Large-scale uplifts in the United States, including those of the Colorado Plateau and the modern-day Appalachian Mountains, are thought to be related to the subducted Farallon plate, which has been moving eastward relative to the overlying continent for the last 100 million years or so. Smaller features like the Illinois basin or Cincinnati arch are explained as lumps and slumps made during the breakup or formation of ancient supercontinents.
How the Word "Epeirogeny" Was Coined
The word epeirogeny was coined by G. K. Gilbert in 1890 (in U.S. Geological Survey Monograph 1, Lake Bonneville) from the scientific Greek: epeiros (mainland) and genesis (birth). However, he was thinking of what held continents above the ocean and held the seafloor below it. That was a puzzle in his day that today we explain as something Gilbert didn't know, namely that the Earth simply has two kinds of crust. Today we accept that simple buoyancy keeps the continents high and the ocean floor low, and no special epeirogenic forces are required.
Bonus: Another little-used "epeiro" word is epeirocratic, referring to a period when global sea levels are low (like today). Its counterpart, describing times when the sea was high and land was scarce, is thalassocratic.