Horse mounts

Horse mounts are largely Alliance only mounts. Horse mounts, in this case, includes those horses that are summoned from another plane (Felsteeds), and those that are embodiments of spirits (Paladin's Chargers, Death Knight's Deathchargers).

Factional mounts: Apprentice
You can purchase these mounts from several vendors: You must have riding skill of 75 (obtainable at level 20). If you are playing as any race other than a human, you will also need to have reputation with Stormwind City.
 * at Theramore Isle in Dustwallow Marsh
 * at the Eastvale Logging Camp in Elwynn Forest
 * outside Old Town in Stormwind City
 * in Menethil Harbor in the Wetlands

Additionally, the Black Stallion mount was only purchasable through Unger Statforth in the Wetlands (now you can buy it from Katie Stokx in Stormwind too). This fact produced many rumours and misconceptions about the mount, despite them, Black Stallion has the same requirements as the other Apprentice level horse mounts. A very common misconception was that the mount was a new mount, when in fact this mount has been available as long as the other mounts. Note there is no riding trainer in Menethil Harbor.

Factional mounts: Journeyman
You can purchase these mounts from Katie Hunter, Katie Stokx or Gregor MacVince, so long as your character has riding skill of 150 (obtainable at level 40). They are reputation restricted like the apprentice mounts.

Class specific mounts
Death Knights, Paladins, and Warlocks are all able to gain mounts specific to their class (and for paladins, faction).

Death Knights
The Death Knight Acherus Deathcharger (ability) is earned by Death Knights upon completion of the quest Quest:Into the Realm of Shadows.

Paladins
Paladins gain the ability to summon a warhorse (apprentice level mount) at level 20, and a Charger (Journeyman level mount) at level 40.

Warlocks
Warlocks gain the ability to summon a Felsteed (apprentice level mount) at level 20, and a Dreadsteed (journeyman level mount) at level 40.

PvP steed
The Black War Steed is available from in Stormwind for Honor points. You do not need to have exalted reputation with Stormwind City to purchase it.

This is a slight variant on an older mount.

Argent Tournament
A number of differnt horses can be purchased from the Argent Tournament, each with different requirements.

The human quartermaster to the Argent Tournament offers the Swift Gray Steed and the Stormwind Steed. You must have completed Champion of Stormwind or be a human to gain access to this quartermaster.

The high elves offer a Quel'dorei Steed through their quartermaster to anyone who is with The Silver Covenant.

Dame Evniki Kapsalis offers the Argent Warhorse to people who have attained the rank of. As noted previously, she also offers the Argent Charger to Paladins with that title.

Horse mounts as loot
A number of horse mounts are available as loot items from various places.

The Fiery Warhorse is looted from Attumen the Huntsman. The mount of the Headless Horseman can be taken from him during Hallow's End.

Horse mounts as quest rewards
The Crimson Deathcharger is a quest reward from Quest:Mograine's Reunion.

Blizzard Pet Store
The Celestial Steed is a supernatural warhorse, the first mount directly purchasable from Blizzard, for $25US or €20/£17, and scales with the player's riding skill and location.

Discontinued mounts
These used the old horse mount graphics and as of Patch 1.4 are no longer obtainable. They can be exchanged for the newer variety if desired, but the players who own them prefer to keep their old mounts due to the fact of their uniqueness and non-availability. Having any of these mounts awards the Feat of Strength Old School Ride.

Recruit-a-friend
The Recruit-A-Friend program rewarded a player with a Swift Zhevra on successfully recruiting someone (who then paid for at least a month's subscription). This has since been replaced with the X-53 Touring Rocket.

Trial of the Grand Crusader
Completing A Tribute to Insanity (10 player) in the Trial of the Grand Crusader offered up a Swift Alliance Steed to alliance 10-man raids (one dropped per raid). Achieving A Tribute to Immortality in 25-man raid offered the Crusader's White Warhorse to the Alliance, and Crusader's Black Warhorse to the Horde.