The Hill sphere, AKA the Roche sphere, is the gravitational sphere of influence of a celestial body. That is, it is the area in which a body's gravity is the dominate force acting on its satellites. By definition, all moons and satellites of a planet lie within their Hill sphere, and likewise all planets of a star.
This is closely related to the idea of a gravity well, but differs in two aspects. First, a gravity well goes right down to the surface of a body, while the Hill sphere refers specifically to the zone in which stable orbits are possible. Secondly, gravity wells can 'nest' while Hill spheres cannot, e.g., the Earth's moon is in the gravity well of both the Earth and the Sun, but only in the Hill sphere of the Earth. It is worth noting that Hill spheres inherently take the movement of the bodies as a given (treating them as a two-body problem); e.g., if the Earth and Moon were to magically stop their orbit of the sun, the Moon would not fall into the Earth, but rather into the Sun. The effects of the Earth's and the Sun's gravity wells balance each other perfectly at ~260,000 km from the Earth, while the Moon is 384,400 km from the Earth. But because the Earth and the Moon are already in orbit around the Sun, the Earth has enough oomph to pull the Moon into orbit around it.
The boundaries of a Hill sphere are somewhat messy; the Lagrangian points are in effect gravitational eddies where competing Hill spheres interact, although most points on the surface of a Hill sphere are simply points where, if a satellite has enough delta-v to pass them, they will slip into the orbit of the next 'competing' gravity well; this might be a satellite of the Earth moving into orbit around the Moon, the Sun, or in some carefully planned cases, any stop on the interplanetary superhighway. Bodies with lots of large satellites -- such as the Sun and its planets --
have spheres that are riddled with 'holes', as each planet and most asteroids have spheres that exist inside the Sun's sphere.
It is also worth noting that the size of a Hill sphere is dependent on nearby bodies. For example, the largest planetary Hill sphere in the solar system is Neptune's. Jupiter has a larger gravity well, but it is nearer the sun, meaning that the sun's gravity has a greater destabilizing effect on bodies in Jupiter's orbit. The effect of the Sun's proximity is so great that Pluto's Hill sphere is larger than Earth's.
Objects in the outer 50%-30% of a body's Hill sphere are often not stable over the long term, and will eventually escape orbit. Other effects, such as passing near other large bodies, radiation pressure, and the Yarkovsky effect can cause objects to slowly escape orbit.
One might make an erroneous connection between the Hill sphere and Hill's Cloud (AKA the Inner Oort Cloud); this is not a reference to being near the edge of the Sun's Hill sphere (it's not, really), but rather it is named after another Hill, Jack G. Hills, who theorized it in 1981.