Questing & Exploration Mounts
Mounts are awarded for a range of different quest activities. In some cases the quests are even designed specifically around the mount - such as raising a mount from a cub or hatchling. Others serve as rewards for reaching the end of a quest chain. Others involve solving riddles or heavy exploration.
Mount Quest Chains
Mounts in this category have quests designed specifically around them - whether it be raising the mount from a youngster (Frostsaber, Ravasaur and Kua'fon), capturing and taming the mount (Garrison Mount Training), or other serialized quests (Falcosaurs).
Mount Quest Chains - Brawler's Guild
These mounts require an extra level of commitment to solving riddles and discovering secrets. Unlike quest mounts, there are rarely any straightforward clues to finding these. Instead, the puzzles surrounding them are usually solved by crowd effort, especially that of the WoW Secret Finding Community on Discord. It would be difficult to solve most of these on your own, but follow their Wowhead links for more detailed instructions.
Zone & Event Quest Chains
Most of these mounts are rewards for reaching significant stages in quest chains.
World / Repeatable Quests
Low-chance drops from repeatable quests.
Kill a crazy amount of NPCs for an insane amount of items and earn a mount.
Mission Board - Draenor Garrison
Mission Boards were introduced in Warlords of Draenor. Two mounts have so far been offered as rare rewards from these.
Other Loot Quest Chains
Mount quests that don't fit in other categories.
Ahead of the Curve
Ahead of the Curve mounts are awarded for killing the final boss of an expansion before the pre-patch of the next expansion hits. Therefore most or all of these mounts are retired.