A flight master is an NPC that offers players means of travel to different parts of a zone via either a Gryphon (Alliance) or a Wyvern (Horde). Flightmasters are commonly found at a roost or nest of bats, gryphons, hippogryphs, wyverns or dragonhawks — depending on the faction and zone a player is in. Although groups of players or level 80 solo players often do so, they are discouraged from attacking the flight masters by making them level 65 elites that spawn two additional aggressive level 65 elite beasts of the appropriate transportation animal when attacked by opposite faction players. Flightmasters don't attack players of the opposite side unless they are themselves attacked nor help you if you are attacked by players of the opposite side.
Some flightmasters, such as several in Hellfire Peninsula, only offer one option to fly and this is typically used for either specific quests, such as bombing runs, or to get to a more centralized location.