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These creature (also known as the Lonely Assassins) are a fictional ancient race of aliens. Steven Moffat, their creator, attributes their appeal to childhood games such as Grandmother's Footsteps and the notion that every statue out there is secretly a Weeping Angel.
Their usual mode of feeding is to send their victims back in time, feeding on the resulting energy difference. When they are not being observed by another sentient being, they can move very quickly and silently, but when they are being observed they become "quantum-locked", occupying a single position in space. In this state, they are frozen and indestructible. They cannot suppress this reaction. If two Weeping Angels were to look at each other at the same time, they would be trapped in stone form until an outside force moves them apart. To prevent this, they often cover their eyes when freezing - this makes them look like they are weeping. They were not shown moving until the Eleventh Doctor episode, "Flesh and Stone", where they slowly turn towards Amy Pond when they realise she cannot open her eyes. They cannot be destroyed.
What is Weeping Angels? Mom, please edit this. I didn't read it very thoroughly as to not give anything away (I have not seen this episode yet)