Reapers were beings linked to Time itself. They appeared at locations of temporal paradox and would "eat" whatever sentient beings they found. The Reapers would search out disturbances in time, such as temporal paradoxes. They then leave the Time Vortex and sterilize the area by devouring everything, with older objects being the tastiest.
Characteristics
Appearance
The Reapers appeared as flying reptilian beings with sleek heads, four long hooked arms and a mouths on their chests.
Behavior
They engulfed creature in a flash of light with their mouth, leaving no traces. Despite these powers, they are material creatures (at least they are in this world), and can be blocked. They cannot pass through solid objects, with older objects being more effective. As damage to Time in the form of time paradoxes accumulated, that allows them entry.
In less than a few hours, Reapers had devoured all but a handful of the Humans of an alternative timeline Earth.
The presence of Reapers would seem more prevalent given, as the Doctor explained, other himself himself, no Time Lords existed to prevent temporal paradoxes in preventing paradox. With the Time Lords gone, paradoxes happened with more frequency. (DW:Father's Day)
The Reapers are similar in both appearance and to the Vortisaurs, the Hunters and the Chronovores. The Doctor did not actually refer to them as Reapers.
Behind the Scenes
The name "Reaper" never actually appears on-screen. Originally, the script for Father's Day described the Reapers as appearing like the traditional Western idea of Death, with a robe and scythes. Presumbably then, they would not eaten their prey but cut them down with their sycthes.