Mounts are awarded for completing an eclectic range of different achievements - from hard-core raiding to collecting toys! The mount reward is often placed in your character's bags or mailed to them in-game, but other times it may be added directly to your account's mount collection.
Miscellaneous Scenario and Instance Achievements
There's a different mount collection achievement for every 50 mounts you add to your collection, and each rewards you with a special mount. Note that the achievement doesn't count every mount on your account. Instead, it counts all mounts usable by one (any) of your characters. So, Alliance-only mounts won't be counted towards the tally on your Horde characters, and vice versa. Class-specific mounts often don't count for either faction.
Rewards for reaching Exalted reputation with various factions.
Awarded for completing a suite of achievements in the 5-player dungeons of an expansion.
Dungeon Keystone & Challenge Modes
In some expansions, a mount is awarded for achieving at least Silver in the Challenge Mode of each 5-player dungeon or 'Keystone Master' since Mythic Dungeons were introduced. These achievements have always been discontinued once the expansion ends.
Generally, there are two or three raid meta-achivements for each expansion, from Wrath of the Lich King onwards. These meta-achievements encompass a range of difficult, fun, and downright insane achievements within the raid. Often one meta-achievement covers all the raids released when the expansion launches, with different meta-achievements for later raid tiers. Older raid achievements are often soloable, but sometimes the mechanics mean that multiple players are needed, regardless of their level.
Ahead of the Curve
Beginning in Mists of Pandaria, Ahead of the Curve mounts are awarded for killing the final boss of an expansion before the pre-patch of the next expansion hits. These mounts cannot be obtained after that point. Only one is listed here because it was the only one awarded directly from the achievement.