Before we start, it is worthwhile to make a distinction between "factions" and "clans" of
Rokugan, the Japanesque setting of the
Legend of the Five Rings card, role playing, and miniatures games. To some extent, the two were once interchangeable terms (in the card game, at least - the paper-and-dice game never had any use for "faction", and I cannot speak for the miniatures), but using current terminology, there is a bit of a difference. This new distinction was instituted to eliminate the confusion caused by clans that weren't "clans" (
the Boar Clan,
the Hare Clan,
the Centipede Clan, etc.) and "clans" that weren't really clans (
the Naga,
the Monks, etc.) "Factions" represent groups with a stronghold -
Toturi's Army,
the Naga and
the Lion Clan would both be considered factions, though the former two do not represent what the average Rokugani would term a "clan", and "clans" represent lineage-based Rokugani institutions, regardless of in-game representation - the term "clan" would encompass
the Tortoise Clan,
the Sparrow Clan and, once again,
the Lion Clan, despite the fact that the former two are not represented as strongholds and a player cannot formally affiliate himself with them. In simple terms, "faction" is a gameplay distinction, and "clan" is a
storyline distinction.
Historically, Rokugan's clan structure began when seven kami, upon falling to earth, founded the seven Great Clans:
The Crab Clan (founded by Hida)
The Crane Clan (Doji)
The Dragon Clan (Togashi)
The Ki-Rin Clan (Shinjo)
The Lion Clan (Akodo)
The Phoenix Clan (Shiba)
The Scorpion Clan (Bayushi)
Meanwhile, Hantei founded the imperial line, Fu Leng landed in the Shadowlands and became corrupted, and Ryoshun was entirely forgotten about until he was needed at the conclusion of the godawful Hidden Emperor story arc.
After a while, Shinjo decided to leave and take a good deal of the Ki-Rin clan with her. They returned a few hundred years later minus Shinjo, plus a few "barbarian" members, most notably the Moto family, and with a new name: The Unicorn Clan. Those former Ki-Rin who never left eventually became the (minor) Fox Clan. While this was all happening, the Phoenix destroyed the (minor) Snake Clan for using maho.
During all this time, various minor clans were being founded. While unlike the Great Clans, they could not claim a kami as their founder, (though the Mantis originated with the son of the fortune Osano-Wo) and did not possess significant might, holdings, or influence at court, they got along, often filling niche roles. This included the Badger, Boar, Centipede, Dragonfly, Falcon, Hare, Mantis, Sparrow, Tortoise, and Wasp clans.
Let's now skip few more years ahead, just around a millennium since the original battle between the Thunders and Fu Leng. For reasons unknown to the rest of Rokugan, the Scorpion Clan decides it would be very wise to march to Otosan Uchi, the imperial capital, kill the emperor, Hantei the 38th, kill his only child, and then hold out against the other clans. They pull off the former two, but not the latter two, and this "Scorpion Clan Coup" is generally regarded as a failure. For their trouble, they get officially disbanded, Bayushi Kachiko, comely wife of the late Scorpion diamyo is wedded to Hantei the 39th, and depending on which version of the storyline you follow, the remaining Scorpions are possibly hunted down.
Over the course of the next few years, all the Black Scrolls are opened, Hantei the 39th falls very ill, and all the clans war against each other. Oops! It turns out that the emperor is possessed by Fu Leng! Oops! It turns out that's what the Scorpions were trying to prevent! Wow! Another Day of Thunder! By this point Yoritomo, daimyo of the Mantis Clan has forged an alliance of minor clans. Through a combination of military prowess, assistance to the Great Clans, and big brass ones, Yoritomo gets the Mantis recognized as a Great Clan. Afterwards, the Scorpions are allowed to reintegrate into Rokugani society, and the Phoenix, hit hard by devastating wartime losses, internal corruption (think less "bribery and nepotism" than "rotting skin and claws"), and Isawa Tsuke's decimation of the Elemental Masters, struggle to rebuild. Meanwhile, for no appreciable reason, the Crab absorb the Falcon Clan.
Just about the time the Scorpions were done unpacking their things, they got framed for the kidnapping of Toturi, the next emperor, and get exiled to the Burning Sands in a poorly orchestrated attempt to boost the sales of that ill-fated spinoff. Eventually (the Ninja impostor) Toturi is revealed, though not before forming the (minor) Monkey Clan for Toku, it's realized that the whole thing is the ninjas' fault, everyone feels bad again, and the Scorpions come back once more. The Kolat help save the world. There have never been any Kolat. Rokugan has always been at war with Eastasia.
While this is all happening, Yoritomo's Alliance began to break down, most notably with the withdrawal of the Fox Clan. Without the backing of the minor clans, the Mantis Clan was weakened, and following the death of Yoritomo himself, in yet another epic battle, the Mantis, while still technically a Great Clan, have been substantially marginalized.
Bringing us up to the storyline present, during the Spirit Wars, Unicorn Clan once-outcast Shinjo Morito founded the Ox clan.