The Roche limit is the distance within which an idealised moon, held together only by gravity, will start to disentegrate due to tidal forces exceeding its gravitational self-attraction. The formula for calculating the Roche limit for a rigid sphere is
Real satellites, both natural and artificial, can usually orbit within their Roche limits because they are held together by forces other than gravitation (primarily the tensile strengths of their materials). Jupiter's moons Adrastea and Metis are examples of natural bodies which are able to hold together despite being within their Roche limits. However, an object resting on the surface of such a body held in place only by that body's gravity could find itself pulled away into orbit by tidal forces.
The Roche limit is sometimes described in terms of "Roche lobes", a three-dimensional mathematical surface which forms a two-lobed figure-eight shape with each of the two masses at the center of its respective lobe. This surface represents a set of points with equal gravitational potential. If you were to hypothetically fill these lobes completely with water a ship could sail from one mass to the other, passing through a point between the two where both planet-wide oceans touch the tips of their sharp peaks together. When an object "exceeds its Roche limit", its surface extends out beyond its Roche lobe and the material which lies beyond "falls off" into the other object's Roche lobe. This can lead to the total disintegration of the object, since reducing its mass causes its Roche lobe to shrink.
Planetary rings are always located within their Roche limit, which prevents their component particles from coalescing into one or more larger bodies. It is theorized that they form when a moon passes within its Roche limit and breaks apart.
The Roche limit is named after Edouard Roche, the astronomer who defined the term.