Reaper

Reapers were beings linked to time itself. They searched for temporal paradoxes. If the balance of time was disrupted, they would be drawn out in force to remove the offending element.

Altering a fixed point in time, or causing the Blinovitch Limitation Effect could draw out the Reapers. (TV: Father's Day, COMIC: Four Doctors) However, if causing the limitation effect or meddling with a fixed point was part of established history, it would not cause a disturbance the Reapers could appear. (TV: Mawdryn Undead, The Wedding of River Song)

Normally, the Time Lords would prevent the appearance of the Reapers and repair the damage to time. (TV: Father's Day) However, following the Last Great Time War, the Time Lords were forced to hide out at the end of the universe to prevent another Time War, (TV: Hell Bent) leaving time more vulnerable to the Reapers as only the Doctor was left to handle repairing the damage. (TV: Father's Day, COMIC: Four Doctors)

Biology
The Reapers were flying gargoyle-like beings with red eyes, two wings, two claws below their wings that could break through stone, mouths on their chests, tails that resembled a scythe and dark brown, tough scaly skin. They lived in the Time Vortex, (TV: Father's Day) as well as the Morass. (COMIC: The Crimson Hand) The Reapers would search out disturbances in time, such as temporal paradoxes, leave the Time Vortex and act "as bacteria taking advantage of the wound" by devouring everything in the area. Older objects were stronger against them and sentient beings were of more interest than inanimate objects. They devoured creatures through their mouths, leaving no trace. They also released a golden glow from their eyes and under their skin when attacking, or about to attack. (TV: Father's Day) Reapers had the ability to deactivate TARDISes as to not allow for an escape from their traps. They could also break their ways into the TARDIS interiors to reach those inside. Reapers could be defeated by making contact with a TARDIS key. (TV: Father's Day, COMIC: Four Doctors)

According to the Ninth Doctor, there was nothing in the entire universe which could harm them. Despite these powers, the Reapers were material creatures, at least in this world, and could be blocked. They could not pass through solid objects. Older objects, such as a church, were more effective barriers. As damage to time in the form of time paradoxes accumulated, it allowed them entry. They could also be banished if the timeline was repaired. If this happened, everything would return to as it was before and memories of the Reapers would disappear and the creatures they killed would be restored to life. (TV: Father's Day)

History
A Reaper roamed the Vortex while Ramón Salamander was drifting through time towards Earth. (COMIC: The Heralds of Destruction)

When the Ninth Doctor brought Rose Tyler to 1987, she saved her father Peter Tyler when he was supposed to die in a car crash. She missed the attempt during her first visit, so the Doctor took her back again. He warned her not to interact with their past selves, but she ignored this warning when she saved her father; a wound in time was created by her intervention, and thus the Reapers were able to enter the universe, and begin "sterilising the wound" by feeding on humans inside it. The Doctor and a handful of survivors managed to take refuge in an old church, where he planned to bring his TARDIS to them. Unfortunately, Rose touched her infant self, allowing a Reaper to enter the church, devour the Doctor and disrupt the TARDIS. The Reapers were expelled from the universe again when Peter restored the timeline by jumping in front of the car which should have killed him. (TV: Father's Day)

Though all trace of them was supposedly erased, the Reaper's intrusion of Earth was in the Atraxi's records. (TV: The Eleventh Hour)

When the Tenth, Eleventh, and Twelfth Doctors all arrived in Paris 1923 at the same cafe, a fight broke out between them, and two of them accidentally crossed fingers. This created a blast of temporal energy, known as the Blinovitch Limitation Effect, which alerted the Reapers to the paradox. The Reapers entered the cafe from the Time Vortex and began attacking the three Doctors and their companions. The Reapers took a strong interest in the Eleventh Doctor over the others, and began chasing after him as the cafe collapsed. Running through the square, he led them past a comic stand which he had visited earlier and to one of the TARDISes parked in the city. This TARDIS had already been disconnected from its interior by the beasts. Using the Blinovitch Effect once more, the three Doctors were able to connect the TARDIS shell to the Twelfth Doctor's TARDIS control room. The Reapers were able to break into the TARDIS, however, and the Doctors began leading the Reaper inbetween the different TARDIS console rooms, dematerialising the ship once they were all inside. This moved them into the Vortex again, meaning that they would either have to leave or die. (COMIC: Four Doctors)

Behind the scenes

 * The name "Reaper" never actually appears on-screen. The script to Father's Day described the Reapers as appearing as multiple Grim Reapers who cut their victims down with their scythe. The idea of a more alien look for the Reapers came out of production team discussions. However, the comic Four Doctors does use the name to identify the creatures. The name "Reaper" is also used for them in Doctor Who: Legacy.
 * A number of people, such as Bee, were said to be late for the wedding, implying they were eaten by the Reapers.