Heroic refers to game content (quests, raids, adventure areas, characters, feats) up to level 20.
Heroic characters cannot enter epic content. At level 20, characters start gain epic powers (mainly from Epic Destinies). Heroic and epic experience is separated, heroic characters cannot gain epic experience until they take level 20 (and stop being heroic). Epic characters can still gain experience from heroic content (though epic quests often yield more XP).
For quests, the quest level on normal difficulty is important for determining whether it's treated as heroic or epic. For example, The Riddle is a level 21 quest on Elite difficulty, however, it's still treated as heroic quest, because on normal it is level 19. Some level 20 quests are treated as heroic (The Dreaming Dark, The Lord of Blades, Tower of Despair), other are epic. In epic quests, depending on the difficulty, monsters gain additional powers, for example, Epic Ward.