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Madame Kovarian was an associate of the religious movement the Silence.
Frustrated by the centuries-long stalemate/conflict between the Eleventh Doctor, the Papal Mainframe and the various enemies of the Doctor during the Siege of Trenzalore, Kovarian and her followers travelled back through time "to train a girl to kill" the Doctor, and prevent the siege from ever occurring.
Some time after Amy's honeymoon, Kovarian arranged her abduction and replacement by a Ganger. Amy's real, pregnant body was taken to Demons Run in the 52nd century. Kovarian watched over her through a hatch above the room Amy was held in. Meanwhile, her linked Ganger lived out her life. This affected the Ganger's mind. She saw visions of Kovarian wherever she went, often hearing her comments on her health and her pregnancy.
The first time Ganger Amy saw Kovarian, she appeared at the little girl's bedroom door at Graystark Hall Orphanage in a hatch in the door. Amy was startled and asked who she was. Kovarian ignored her and said "No, I think she's just dreaming", then disappeared. The room was empty, with no evidence of Kovarian or the hatch.
While sleeping on a pirate ship, Amy woke to see Kovarian in a hatch that opened on the hull. She said, "It's fine. You're doing fine. Just stay calm." When Amy looked away, she vanished along with the hatch.
She was seen twice by Ganger Amy at an acid mining factory, first whilst Amy searched for Rory, though she did not speak. The next time, Amy sensed her coming appearance and stared at the wall where the hatch would open. When she mentioned this to the Ganger Doctor, who was pretending to be the Eleventh Doctor at the time, he dismissed it as "time memory, like a mirage." After her Ganger form was destroyed, Amy awoke to find herself in an alien hospital and heavily pregnant, with Kovarian looking over her. She went into labour, with Kovarian cruelly instructing her to push.
Madame Kovarian and Colonel Manton visited Dorium Maldovar to ask him what he knows; they had been on Yellow Alert for a month and the Doctor hadn't done anything. Dorium explained that there were numerous people throughout time and space who owe the Doctor a debt for helping them; he's gathering an army. Dorium explained the stories about the Doctor were true and not myths. Seeing they were getting nowhere, Kovarian and Manton left.
At some point after Amy's baby was delivered, Kovarian took her away and had a Flesh duplicate created as a decoy for the Doctor. The duplicate and Amy were "rescued" by Rory Williams. During the battle with the Headless monks, Kovarian told the Doctor she was "privileged" to have fooled him twice in the same manner and disconnected the Flesh signal, disintegrating the duplicate.
She appeared at the Luna University circa 5123 to River Song. Kovarian taunted her over her capture as a baby and her role in the Doctor's death. With Silence and soldiers from the Church, she took River and forced her into an astronaut suit with an upgraded weapons system which controlled her nervous system. River was taken to 22 April 2011 and submerged in Lake Silencio to lie in wait to kill the Doctor.
Kovarian was in the Area 52 pyramid in the alternate timeline created by River Song's refusal to kill the Doctor at Lake Silencio, having been captured by River and Amy Pond's military group and tied to a chair. They had created their own Eye Drives based on hers to remember the Silence, but as she told the Eleventh Doctor and his companions, the drives gave an advantage to the Silence; they could electrify them remotely, causing debilitating pain or death. The Silence began electrifying the Eye Drives, including Kovarian's. She pleaded with Amy to save her, but Amy took revenge on her for the abduction of her baby by forcibly reinserting her Eye Drive and leaving her to die. Amy remembered this after that reality disintegrated and was conflicted over her actions, although River tried to reassure her that it was an aborted timeline and didn't count.