For a moon in orbit around a planet, there comes a point at which the gravitational force of attraction of the nearest side of the moon to the planet is so much greater than the gravitational force of attraction of the far side of the moon that the moon gets ripped apart.
We call these differential gravitational forces tidal forces.
Click here for a visual representation of the effect — repeatedly click the right-arrow to animate the images.
Jupiter’s closest moon, Io, is outside the Roche limit, but nonetheless experiences significant tidal forces, causing large amounts of internal heating. This is the reason why Io is the most volcanically active body in the solar system and explains its relatively smooth surface (which is constantly being refreshed with new molten rock) and yellow colour (which comes from the sulphur in volcanic eruptions).
Mars’s closest moon, Phobos, is spiralling very slowly towards the Roche limit. In around 100 million years time, it will be ripped apart by tidal forces.