Horselord

The Horselord is a Barbarian Variant Class from Dungeons & Dragons 3rd Edition that debuted in Dragon Magazine #338. It's basically a mounted barbariant variant, inspired by the Mongols and to a lesser extent by the horseriding warriors of the Native American tribes from classic Westerns. Mechanically, horselords are identical to standard barbarians, except as follows:

• They lose the Fast Movement and Trap Sense abilities.
• They gain the Expert Rider ability; they suffer no penalties to Ride checks for riding bareback and gain a +2 bonus to Ride checks when using a saddle. They also gain Mount Familiarity, which gives them a +2 bonus to Handle Animal checks with a single species of rideable animal - this is most conventionally a horse, but could be anything your setting allows to be ridden, even weird fantasy steeds like chocobos and dinosaurs.
• They replace Rage with Battle Ecstasy, which functions largely the same; when a horselord enters Battle Ecstasy, they gain +4 Dexterity, +4 Constitution, and Damage Reduction as per Rage. A horselord can use any skill or feat whilst in Battle Ecstasy, but suffers a -2 penalty to all of them except for Handle Animal and Ride. Battle Ecstasy lasts as long as Rage and fatigues the horselord just like a rage, until they reach 17th level when the fatigue penalty vanishes. There is no Greater Rage or Mighty Rage equivalent for Battle Ecstasy.
• At 4th level, they gain Faithful Steed; essentially, after spending 2 months owning and riding a specific mount, the horselord can turn that mount into an Animal Companion, which functions identically to the animal companion of a ranger of the same level, except that its type remains Animal, not Magical Beast. Replacing a slain Faithful Steed requires 1d4 months of searching and a successful DC 15 Handle Animal check.
• Finally, at 6th level, the Horselord gains Improved Mounted Archery as a bonus feat.

Pathfinder

There's also the Horse Lord archetype for the Ranger class that Pathfinder made for the Advanced Player's Guide splatbook, which can make things confusing if you're trying to keep track of everything. Their focus on mounted combat is pretty much identical, but it's built more around making that mount good rather than throwing the mount into another class's benefits.

• They get a class feature called "Mounted Bond" at level 4, which gives them an Animal Companion in the form of a creature they can ride. They can only ride horses or camels, with a provision for smaller riders being able to get away with ponies and wolves as well as boars and dogs at level 8. This pretty much locks you to the companion path instead of the shared Preferred Enemy bonus. Considering how incredible horses are, this is hardly a loss. That said, it's noted that this ability "functions as the Druid Animal Companion, with the ranger treating their effective Druid level as (Ranger Levels - 3)", so theoretically this could be used to add more fantastical steeds as you gain levels as well as prestige classes.
• Loses Camouflage and Hide in Plain Sight, which is just as well because horses aren't known for being stealthy. The former is excellent by removing the level penalty given to animal companions, but it comes online quite late. The latter is more icing, as it gives the pet THP once per day for some extra protection.
• Locks the Fighting Style option to the Mounted style. It gives a couple decent and necessary feats to pick up, but it all comes at the cost of being able to take feats more vital in optimizing their offensive output.

However, the idea of a mounted barbarian also survived into Pathfinder 1e, and even made it into the same splatbook as the new Horse Lord. This archetype is called the Mounted Fury, and the shared roots are obvious:

• Trades Fast Movement for Fast Rider, adding that speed boost to the mount instead of the Barbarian.
• Trades Uncanny Dodge and Improved Uncanny Dodge for Bestial Mount, which is the same as the Horse Lord's Mounted Bond.
  • This may not seem like much, but the APG also adds a new array of rage powers for mounted barbarians, some of which were specifically recommended to the Mounted Fury - Ferocious Mount (regular & greater; shares the benefits of Rage to mount except the damage reduction at the cost of an extra round of rage), Ferocious Trample (regular & greater; adds the Trample action to the mount), and Spirit Steed (Gives damage reduction when owner and mount rage), specifically.