The Roche limit (or Roche radius) is the nearest to an astronomical body that a particular body orbiting it can approach before breaking up due to tidal forces. A moon orbiting a planet within the planet's Roche limit for that moon is being pulled apart by the planet's gravity more than its own gravity is holding it together. Thus at this limit, moons tend to break up into rings like Saturn's, or planets orbiting stars tend to break up into circumstellar disks. The exact formula depends upon a few factors (including the masses of both objects), but a basic formula is:
d = Rm(2 MM/Mm)1/3
This formula is derived by determining the distance at which the gravitational force and tidal force balance. If a longer distance apart, the secondary body's gravity dominates throughout its own radius, holding it together, and if a shorter distance apart, the tidal force from the larger body overcomes its the secondary body's gravity and pulls it apart.
The term Roche sphere is generally used as a synonym for Hill sphere, which applies to the influence of the host's gravity as if tidal forces were not a factor: the Roche/Hill sphere's radius is not this Roche radius, but is termed the Hill radius.
The term Roche lobe seems more related to the Hill radius than to the Roche limit, i.e., the Roche lobe is not defined by tidal force: it indicates the actual elongated (non-spherical) region of an orbiting body's gravitational influence. The Roche lobes of each of two co-orbiting bodies meet at their L1 Lagrangian point.