Quests are divided in several categories, ranging from "Main Scenario" quests, which are the most involved and very cutscene-intensive, to the simpler "Sidequests", that are mostly small fetch quests that offer some insight on the game's lore and offer gil rewards. Unlike Guildleves, quests cannot be repeated.
Some elements and areas of the game are locked until certain quests are completed. For example, players cannot own personal chocobos before enlisting on a Grand Company, create materia, or align with friendly sects of Beast Tribes before doing the appropriate sidequest.
Missions have a multitude of objectives, some are as simple as just traveling to a certain area, while others might involve parley, crafting or battlefields with unique Notorious Monsters, for example.
Missions and Quests
Shown below are the quest types available in Final Fantasy XIV: