A Mongol without a horse is like a Bird without the wings
-Traditional Mongolian saying
A culture so thoroughly in love with horses, they may as well be centaurs. They fight on horseback, they travel on horseback, they eat on horseback, they sleep on horseback. They probably aren't literally born there, but they'll certainly start learning to ride before they can walk.
Expect a lot of jokes about just how much they love their horses.
Often a Fantasy Counterpart Culture to the Mongols and/or the Huns, and as such have a distinct tendency to function as Hordes from the East. However, as the Lord of the Rings example of the Rohirrim demonstrates, they can just as well be a more civilised, chivalric culture. Whether or not they're the bad guys, they're sure to be a Proud Warrior Race (you can't get much studying done at a canter, after all).
Usually expert Horse Archers, though tend to be just as good with a sword.
Sometimes the steed of choice will be a Horse of a Different Color, but actual Equus f. caballus seem to be favoured for some reason.
The Dothraki in A Song of Ice and Fire don't let you forget about it. They worship horse-gods, and talk about practically everything in terms of equine metaphors. And changing the word from Khan to Khal isn't fooling anyone, either. They even refer to what we call "doggy-style" as "horse-fashion," and most will have sex in no other way.
The Horse Tribes in The Light Fantastic. They aren't very sentimental about their steeds, though; when your entire culture is based on horses, that includes your diet.
The Algar from The Belgariad. On rare occasion, they actually have members born to the tribe who can talk to horses. They're also cattlemen; when Hettar mentions his dream of domesticating and/or crossbreeding a type of flesh-eating wild horse, Garion (after noting Hettar can't stand the idea of something equine that he can't ride) points out that it's a bad idea to have horses that see cattle as food when your economy revolves around cattle.
The MacAhern clan from the The Witches Of Eileanan, who aren't called "The Horse Lairds" for nothing. They ride winged horses; supposedly, you have to stay on one for an entire year, day and night, doing everything in the saddle before the horse will accept you.
On Gor the Wagon Peoples, natives of the Tahari region, and Red Savages are all described like this at one point.
The Shin'a'in in the Heralds of Valdemar series, and the Heralds themselves after rigorous training (though their mounts are intelligent spirits in horse-like physical bodies). When Talia is brought to the Heralds' Collegium, her equestrian instructor Keren informs the girl that "by the time I'm done with you, anything you can do afoot, you'll be able to do from the back of your Companion." Later in the trilogy, that same instructor manages to slingshot into a dive from the back of her Companion in full gallop. Said one of the other Heralds, "I didn't know you could do that!"
In The Wheel of Time, this is the Saldaeans' hat. At one point a legion of them parade across a plain doing tricks- riding two horses at once with one foot on either saddle, slapping the ground in passing while dangling precariously from the saddle, or climbing underneath the horse and up the other side, all while at a full gallop.
Basic Dungeons & Dragons, Gazetteer 12 The Golden Khan of Ethengar. The Ethengar tribesmen (who are based on the Real Life Mongol hordes of Genghis Khan) are described this way. Children start riding at the age of three.
The Valenar elves of Eberron. The Halflings of the Talenta Plains also, though they use dinosaurs for mounts.
The people of Nova Vaasa, in the Ravenloft setting, track their origins to nomads Born in the Saddle, and are still renowned as the Core's best horsemen and horse-breeders. Actually an aversion, as this is "false history"; like most Core realms, Nova Vaasa was created out of whole cloth, backstory included, by the Dark Powers.
The Ungols of Kislev, although not quite as formidable as their Kurgan cousins.
The people of the duchy of Couronne in Bretonnia. All Bretonnian nobles love horses, but these guys take it to extremes. Couronne is also the only place in Bretonnia where non-nobles are allowed to ride.
Hobgoblins in the East, although in their case it's wolves.
In Warhammer 40,000, the barbaric Atillan Rough Riders supply some of the most celebrated cavalry regiments of the Imperial Guard (Mogul Kamir even has a literal Automaton Horse). The denizens of the planet Chogoris have a similar mounted culture, though if they're found worthy of joining the White Scars space marine chapter they'll exchange their horse for an armored motorcycle.
The Marukan Alliance consists of this, with a bit of cowboy thrown in for good measure. They worship a horse god (who sires some of their steeds), their national militia doubles up as a postal service, and they often serve as elite cavalry for the nearby city-state of Lookshy.
True to their real-life reputation, the Mongols in the Civilization series tend to get a powerful mounted Unique Unit in the early game.
In The Gamers Alliance, the Khitans from the Khitan Khanate in Maar Sul are expert riders who are skilled in the use of both the bow and the sword. They bond with their horses at an early age and see them as their brothers and sisters.
It is said that Mongolian man wasn't much of a man if he didn't have a horse.
Juan-Juan, Xiongnu, Jurchen, Khitans, basically anyone living north/northwest of the Chinese Empire were almost predominantly proud horse people. Their raids proved destructive for much of Chinese history, giving rise to a massive linked system of forts and walls that eventually became the Great Wall.
Turks and Turkic people. Part of the reason why Greeks used to be so angry with the Turkish people (apart from invading Greece) was the fact that the Eastern Roman/Byzantine Empire (which was predominantly Greek) fell to a people that initially were horse nomads.
The Cossacks of Ukraine and Southern Russia could allegedly ride a horse at the age of 3.
The Scythian people, whose reputation saw huge numbers of them serving as cavalrymen to many Greek and Persian kingdoms during the classical era.
Rivals of the Scythians in the plains of Eurasia, the Sauromatae/Sarmatian people also have quite a reputation as horsemen, that Romans employed boatloads of them for their cavalry, promising Roman citizenship in return for their services.
Some historians think that the original King Arthur was a Roman-Sarmatian cavalry officer. The nobility of the Polish-Lithuanian Commonwealth (themselves renowned for the quality of their cavalry, but not quite to the extent of this trope) also believed themselves to be descended from the Sarmatians, and this view had a major influence on their culture.
The Sioux, Apache, or any Native American nation of the Great Plains. Compared to other horse nomads they spent much of history not having any horses until European explorers brought some over during the 1500's. Being plains people, they quickly adapted to horsemanship and became great riders & cavalrymen by their own.
The Huns. Attila is said to have learnt to ride before he could walk.
Though not much of a national/ethnic entity, cowboys would often love to think they are this trope.