A non-player country (NPC) is, broadly speaking, any map entity under the responsibility of the game moderator. This includes sovereign states, rebel groups, and anything else that can be tracked on the map.
NPCs originated as the result either of players abandoning the game or taking temporary leave, their countries given over to GM custody. They were generally left alone by other players and so did not meaningfully impact the game. Imperium Offtopicum IV was the first game to codify NPC roles, introducing them as minor, indistinct, slowly-expanding nations to act as buffer states between larger powers; they were not integral to diplomacy, and only Mad Man attempted to conquer one.
NPC roles have varied widely in games since, but generally abide by the principle that they are secondary to player nations and player diplomacy, although they can contribute meaningfully to strategy depending on how they are implemented. A notable divergence is Taniciusfox's Sons of Mars series, which emphasized the application of soft power to manipulate NPCs which were given roughly equal footing to player states. Taniciusfox's approach has been criticized in subsequent games for de-emphasizing the players' role; he has been accused of 'guiding' NPC behaviour to elicit a specific outcome, admitting that "My GMing is Keynesian. If gameplay is in a downturn, I increase my activities to stimulate it."
In addition to their roles as minor states, NPCs can also act as unaligned insurgent groups. The question of who controls rebels arose heatedly during Imperium Offtopicum III as a result of a roleplay-generated civil war in the Byzantine Republic. When Mathalamus, realizing his "allies'" intervention was motivated by a territory grab, moved to instantaneously dissolve the rebel armies, other players protested, arguing it was yet another instance of his powergaming and demanding Mad Man disallow it and/or take control of the rebel group himself. Roleplay-based rebellions have since been rare to manifest on the game map lest a player actually lose total control, with GM-run rebels typically created through set game mechanics.
Beginning in the Multipolarity series, Taniciusfox began a practice of two-tiered NPCs. Whereas "minor" NPCs are subject to arbitrary and/or statutory inhibitions (the client mechanic being the most recognizable), "major" NPCs enjoy the full powers of player countries, and aggressively pursue diplomatic, economic and military objectives to comparable competence. While this helps preserve the standing of countries whose original players quit the game, it has been criticized for eroding player agency: the Italy-Russia Alliance and HITSGDP provide examples of major international organizations created solely through NPC initiative.