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 Ultimate Combat splat, 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 can only ride horses or camels, with a provision for smaller riders being able to get away with ponies and wolves. 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.
  • 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.