Nova Terra Interregnum
Nova Terra Interregnum is one of the major time periods of the Imperium of Man stretching from mid-M34 to the beginning of M36. It was preceded by the Forging (M32-M34) and followed by the Age of Apostasy (M36-M37).
During this time period, the leaders of Nova Terra declared themselves independent of the High Lords of Terra, and the Imperium was split in twain, though in reality it would have been a three way struggle politically between the Secessionists, the High Lords, and the Ecclesiarchy, with the latter two being nominally unified against the separatists.
Despite the fact this sounds Awesome, there is little fluff around it, mostly because it doesn't have anything to do with most of the current major factions, besides the Ecclesiarchy. So most of what we know has been pieced from several sources.
It should be pointed out that the framework for the Imperium was pretty poor during this time and looked nothing like the 40k Imperium. The Ecclesiarchy wasn't a unified whole, but the largest faction consisted of the Temple of the Saviour Emperor which had a seat on the High Lords of Terra and was leeching power and tax money from the Administratum (eventually leading into the Age of Apostasy).
One example of the idiocy of this time is that the Ecclesiarchy nearly bankrupted the Imperium by moving the seat of power from Terra to Ophelia VII and then back again, just to prove a point to the Administratum that it shouldn't fuck with the church (hilariously, this is based on a real incident in the history of the Roman Catholic Church); this really should've gotten them executed but oh well. During that time of separation the church had also built on their Frateris Templar to the point that it now supposedly rivalled the size of the Imperial Guard.
So while the High Lords of Terra might have started out by echoing the original Council of Terra which had managed bureaucracy for the Great Crusade and had presumably stayed there following the Emperor's enthronement following the Horus Heresy, it was now in a worse state than it would become by the current era since religion was even more relevant and the actual government had very little power. Since the Emperor approved the creation of the Imperial Senate, this would make everyone who favored the Ecclesiarchy's power over the High Lords' power heretics.
However, it's not clear why the High Lords didn't just murderize any group that gained too much illegal power and request Space Marines enforce Compliance; it would have been easy to point out to the masses that only the Custodes know the Emperor's true will and that the Emperor created the High Lords whereas the Ecclesiarchy was just some dudes claiming the Emperor chose them despite the High Lords having actual proof of their own Emperor-given power.
Similarly, the Inquisition was concerning itself with mostly Daemons, Aliens and rogue Assassins, since the Ordo Hereticus wasn't invented yet. So the concept of religious "freedom" also goes hand-in-hand with excess.
The Main Event
When someone on the planet Nova-Terra had the wise idea to split from the Imperium and form their own council in the Segmentum Pacificus, we can sympathize with the idea of separating from all of the religious furore spouting from Terra and establishing a government that was of a more appropriate "local" scale. It seems that they didn't want to truly leave the Imperium but rather sever the High Lords' authority from themselves but otherwise remain loyal to the Imperium (like what Huron Blackheart tried, with similar results).
But this writing comes from GW so anything resembling common sense has to be purged with fire - and that's pretty much exactly what happened, though the Ur-Council made a good shot of it and ruled the segmentum for over 900 years. We can infer that the Ur-Council was an attempt in the Pacificus to establish a Council of Terra with actual power, given the name of the planet chosen to capital them was Nova Terra and all and the council's own name basically means "original council" ('ur' means 'primeval/proto') or something like that. Most likely they believed they had to make a new Council of Terra without the Ecclesiarchy's control as the Emperor was the one who said the Council of Terra should rule the Imperium in his stead. Apparently obeying the Emperor is Heresy.
Most of what we know about the Ur-Council tends to be in fluff surrounding the Dark Angels and there are strong hints that Cypher, supported by other Fallen Angels, was one of the Ur-Council's leaders and the Dark Angels were particularly active during this period. This is also where the Praetor launcher came from.
The Cataclysm of Souls
An awesome-sounding but sadly limited piece of fluff, the Cataclysm of Souls was a period of religious wars at the end of M35 when the Ecclesiarchy reshaped the Imperium as a religious state, marking the end of the Nova Terra Interregnum. It is not entirely known what happened during this period though it is remarked that it was a time of civil war the likes of which had not been encountered since the Horus Heresy.
The reason the religious wars of the Imperium got the "religion" put in it is because the Emperor set the tone for how the Imperium enforced official religious policy: violent purging of non-believers. So He invented true religious war as a result of trying to end non-existent true religious war through the atheistic version of religious war.
Side Show: Moirae Schism
Not only was the Ecclesiarchy attempting to supplant common sense with religious blindness during this period, but the Adeptus Mechanicus was having its own problems too. Somewhere on the Forge World of Moirae in the galactic North, someone remembered that the Mechanicum were supposed to be worshiping the Emperor in his aspect as the Omnissiah and started making "radical" assertions that the Cult Mechanicus and the Ecclesiarchy should be unified into a single whole, even presenting mathematical evidence based on fluctuations on the energies given out by the Astronomican which were purported to predict the future. They were right.
In a different time, the Mechanicus would have squashed this sub-cult as heretical, but because the Imperium was itself in the grip of religious hysteria surrounding the Emperor, the idea actually took root and spread throughout the galaxy, even among some Space Marine Chapters such as the Iron Hands. Eventually this ideology would spark its own civil war, the first casualty of which was the world of Moirae itself, which was bombed to oblivion by the Martians, the Mechanicus once again showing their trademark directness when faced with a problem.
The conflict would last for over a millennium - actually longer than the Nova Terra Interregnum itself, bleeding over into the Age of Apostasy and only ending slightly before the beginning of the Age of Redemption (during the Thorian Reformation restoring the High Lords of Terra after Goge Vandire's reign of terror). It is not clear how it ended other than that many of the radicals were brought back into the fold of the "core" Cult Mechanicus. With the Iron Hands and their successors, the separatists were permitted to go their own way and form their own separate chapter named the Sons of Medusa, which was specifically ratified by decree by the High Lords and is officially not a part of any founding. This was not the case for the Red Talons, who purged their Moriae dissidents to the last.
Immediate Aftermath: Cursed Founding
The 21st Founding happened shortly after the Nova Terra Interregnum and was supposedly the largest founding since the Second Founding. Unfortunately those created by this Founding were wracked with awesome mutations and the majority of Chapters created in this founding got wiped out by the Grey Knights.
While the Inquisition and the Grey Knights did start going around offing any the 21st founding chapters that went over the line with mutations (Flame Falcons for example, who were literally on fire all time) or actual treason, others escaped or have toed the line enough they are tolerated by the Inquisition (if only barely). The Blood Gorgons turned traitor and fled to hook up with Chaos while the Black Dragons, Minotaurs, Lamenters, and Sons of Antaeus are all still on active service. The Fire Hawks went missing in action and would later (or possibly earlier, thanks to some warp-fuckery) become the Legion of the Damned.