They're basically a super secret, super-duper elite unit that's known only to the Exarch (until they're unleashed), and even the Exarch doesn't seem to know all the details, unless told by a former Exarch.
Like the Clans, they're trained from an early childhood, but unlike the Clans, they're trained in multiple types of units. They start out as Technicians, then go to Infantry, Armored Infantry Vehicles (called Jaegers, IIRC), then MechWarrior. After that they're allowed to use whatever they want when in combat.
Their largest mentioned unit is a Century, which is 100 warriors (or units, I'm not entirely sure on that from SyD) plus training auxiliaries, which is split into 2 Battlegroups, which are split into 3 Umbras. Each Umbra is supposed to be about equivalent to a mixed company.
Their background is so mysterious that they have implants that others can trigger that immediately destroy all traces of DNA so if they fall on the battlefield their genetics can't be traced. But not so super-secret that their icon doesn't have a Smoke Jaguar on it.
They feel that they owe the Republic their Service for Stone's actions during the Jihad, and that's why they fight for them. They're extremely pragmatic, seeing the mission as the end all, be all, so they're perfectly willing to do whatever it takes to succeed in their mission, including blowing up full hospitals to drop them on the enemy. They're also extremely antagonistic to other Clanners, committing warcrimes such as chopping the heads off of ejected pilots with machetes.
Equipment wise they seem to have whatever they want, including a WarShip. Their production facilities are unknown, but they've basically got an island to themselves on New Earth. Due to their elite technical abilities, they have a habit of constantly modifying their gear based on their own fighting style and the needs of the mission.