The Silence, also known as Is That The Fucking Slender Man?, were a religious order or organization centered around The First Meme Question (lol "doctor who" was said on the show so funny) never being answered. They had priests (which were not that bad like other priests, but were kinda scary I guess) that had the power to make you forget about them instantly after seeing them, like you wish you could do with Kill the Moon. They could be a church wanting to revive Omega or something cool like that but instead we get a group of villains who have a confusing objective only explained in one sentence in Time of The Doctor, whose only purpose is to cause everyone to say "Doctor Who" every single episode, because the 50th was coming and everyone was really proud of Doctor Who at the time. Everyone regrets it immensely.
The Church of the Papal Mainframe enforces a truce around Trenzalore in an attempt to assist the Doctor in containing the situation. It is made clear that if the Doctor speaks his name, the Time Lords will open up a crack in the universe and bring Gallifrey back, and then the entire universe will descend upon them and start a brand new Time War. Therefore, the church of the Papal Mainframe changes its faith and becomes the Church of Silence. On the fields of Trenzalore, at the Fall of the eleventh, when no living thing may lie or fail to answer, the question will be asked and silence must fall.
One part of the church, the Kovarian chapter, broke off from the church of the Silence and went radical. They chose to approach the situation proactively and thus prevent the Doctor from having ever reached Trenzalore to begin with. To that end, they captured his companion, who'd become pregnant while traveling in the Time Vortex, and replaced her with a ganger duplicate. The ganger was drawn into a series of adventures during this time, with the Doctor slowly becoming aware that his companion had been kidnapped.
During and after her pregnancy, the Kovarian chapter performed experiments on her daughter to turn her exposure to the time vortex into full-blown Time Lord-hood. They then sent this child, Melody Pond, to live in an orphanage in America, tended to by the genetically engineered priests of the Silence, who had been on the Earth for thousands of years (since the wheel and fire, at least) driving the human race to the stars so that they would have a power armor space suit to put Melody Pond into. This was pointless because they already came from the future with real power armor and real space weapons to use.
The Silence relocated Melody Pond from the spooky orphanage she lived in to a warehouse in Florida, built overtop a proto-TARDIS designed and built by the Silence. The process of being put inside the space suit was so traumatic for Melody Pond that she called up the president of the United States to try and get rescued from it, but it was pointless because she ripped her way free anyways. The president sent Canton Everett Delaware III, the Doctor, Amy and Rory Pond, and Melody Pond to investigate Melody Pond's phone calls. Melody Pond, by now going by "River Song", was pretending not to know what was going on (to avoid causing a paradox), and thus was already actively sabotaging the plan to kill the Doctor.
It's possible that this was all part of the plan, because it was during these events that Amy Pond's Ganger was captured by the Silence, tests were conducted on it over the course of a few days, and she was also told to inform the Doctor that she was pregnant, thus leading to the events of her rescue from the Silence by the Doctor. The remainder of the adventure resulted in the Silence losing their foothold in the 20th century when the Doctor used their own powers against them to force them into exile from all human-controlled territory. During this battle, their proto-TARDIS was lost in time and eventually landed above Craig Owens' apartment in Colchester, 2010.
As a result of this exile, the Silence were unable to recapture Melody Pond and put her back in the space suit. The half-Time Lord child toddled from Florida to New York, where she regenerated in an alleyway in early 1970. 19 years later, she'd aged 8 years and managed to toddle across the Atlantic ocean, where she enrolled herself in Leadworth elementary school and grew up with her parents, making sure they would grow up and fall in love in order to assure her own conception aboard the TARDIS thus giving herself half Time Lord powers, which she needed for reasons left unclear.
After the Doctor rescued Amy Pond from the Kovarian chapter of the Silence in the future, he went on a journey to try and find out where Melody Pond had been taken. We're forced to assume that by the time he caught back up with them during the events of Miracle Day that he'd figured out she was the girl in the useless power armor space suit, but hadn't figured out that she'd gotten lost in New York somehow and ended up growing up with her parents.
Amy decided to name her daughter after her layabout daughter who was always getting into trouble. The Doctor visited the Ponds to give them a status update on his search for Melody, met Melody, and took her on an adventure to Berlin in the late 1930s at gunpoint. Rather than shoot him with a pistol twice while he was there in the field, or in his TARDIS, or immediately upon arrival in Berlin, or any of the times that he'd returned to see the Ponds during her entire history with them up to that point, Melody instead chose to try and assassinate the Doctor immediately after regenerating into the form most commonly known as River Song.
She first tried to shoot him with her pistol, then with Hitler's pistol, then poisoned him with lipstick made from the poison of the Judas tree, poison that could shut down the ability of a Time Lord to regenerate. It was nearly fatal, but by this point the Doctor had become so attached to River Song's future self that his pleas alone was all it took to get even the past version of herself to help him heal from the poison, thus removing her future ability to regenerate and further sabotaging every future attempt by the Kovarian chapter to kill the Doctor.
After attempting to assassinate the Doctor and failing, Melody Pond is sent to a hospital in the future. After that, she becomes an archeologist with the intention of finding the Doctor, but her adventures come to an end when the Silence again recapture her, putting her back in the useless power armor space suit from the 1960s that can move on its own so she doesn't even need to be inside it and in fact can sabotage its efforts from within it. She was then submerged in Lake Silencio, Utah in April of 2011, a still point which would make it easy to cause the Doctor's death to be a fixed point in time.
The Doctor went to Lake Silencio willingly after ages of dillydallying. During this dilly-dallying, he met the Tesselecta while it was investigating the church of the Silence (who, for reasons unknown, had kept their entirely noble motives a secret) and planned to use it to fake his death. He further blackmailed another member of the Silence into helping him uncover a bit about their motives, namely the central mythos of their church's obsession with the Doctor's silence falling. Before he arrived at Lake Silencio in what appeared to be a rented station wagon, he invited Amy and Rory Pond, a future version of their daughter Melody, Canton Everett Delaware III, and a younger version of himself to get caught up in events.
River attempted to stop the second assassination attempt from happening, either unaware or unwilling to accept that doing so would break one end of a stable time loop and thus cause a massive time paradox, exacerbated by her being present at the events three times (twice as Melody Pond's third incarnation, once as a ganger fetus) and the Doctor being present twice. The resulting time crash nearly destroyed the universe, and during it Madam Kovarian herself was seemingly punished severely by the priests for her terrible plans.
The time crash was resolved when Melody Pond realized that the Doctor was riding inside the Tesselecta, thus allowing him to fake his own death and make the faking of it a fixed point in time even the Silence couldn't alter. Melody Pond was taken to the future by... someone and sentenced to spend seemingly the rest of time in Stormcage prison, where she would regularly break out and go on adventures with the Doctor.
Once such adventure took place during the last attempt by the Kovarian chapter of the church of Silence to assassinate the Doctor. Melody broke out of prison when Winston Churchill called her and told her to break into the national archives and find the Van Gogh painting entitled "The Pandorica Opens", which was influenced by seemingly psychic dreams Van Gogh was having of events that were going to have happened in 102 AD.
These events were the result of a conspiracy arranged by an alliance of alien races, COINCIDENTALLY THE SAME ONES THE SILENCE HAD FORMED TO PROTECT THE DOCTOR FROM, all converging on Britain in 102 AD and laying the seeds of mysterious events around the universe, using imagery drawn from the mind of his companion, all in order to lure the Doctor there and lock him inside the Pandorica prison. Thus, he could not stop the Kovarian chapter from blowing up his TARDIS. The exact mechanism by which this was achieved has never been stated and at this point it's safe to assume it never will be, so my best bet is Melody Pond, their ultimate perfect agent, did it herself while flying the TARDIS (after it had taught her how to do so in the future from its perspective and the past from hers) after her "Silence will fall" programming kicked in. The Doctor and all his companions were effectively dead and the universe was destroyed, but the resulting explosion being poured through the vortex at all moments in space and time caused the universe to be retroactively cracked and temporarily destroyed.
These actions caused the structural weakness in Trenzalore that the Time Lords were calling through, thus creating the very situation that called for Silence in the first place. Through these actions, the Kovarian chapter of the Silence created the very impetus for its own existence, regularly sabotaged its every plan, and throttled their every plot right in the cradle. Melody Pond went on to become an archeologist, still meeting the Doctor out of order, until one day she met the tenth Doctor before he ever met her and died as a result, was uploaded into a computer, and then visited one possible future version of Trenzalore where other stuff happened that had nothing to do with the main plot.
Madam Kovarian made two significant appearances in series 6 and was never seen again. The eventual revelation of the Silence undermined everything interesting about them and turned one of the eleventh Doctor's coolest moments into a pointless act of self-sabotage. The Kovarian Chapter of the Church of Silence can and *must* go down in history as one of the worst villain conspiracies ever conducted on the program.