Lagrange points were discovered by the Italian-French mathematician Joseph Lagrange. Lagrange points come about by solving a restricted three body problem, where one of the bodies has much, much smaller mass than the other two. An example of such a system would be the Earth-Sun-spacecraft (such as SOHO) system.
Such a system must be in rotation for Lagrange points to exist. If the system is not rotating, point L1 will exist for as long as it takes the two large bodies to fall into each other, but all of the other points require the two large bodies to be in orbit around one another.
A given three body system has five Lagrange points. Three of these points lie on the line formed by drawing a straight line between the two larger bodies. The most well know Lagrange point, L1, lies on this line between the the two large bodies. At point L1, the gravitational forces acting on a small body, such as a spacecraft, due to the other two bodies balance out. If you visualize a plane,
perpendicular to a line drawn between the two larger bodies (the Earth-Sun line), you can image a spacecraft being in orbit around the Earth-Sun line with the spacecraft's orbital path lying in this plane. Since the Earth is orbiting the Sun, this plane will change its orientation as the Earth moves around the Sun, to stay perpendicular to the Earth-Sun line. A spacecraft attempting to orbit in this plane about the Earth-Sun line will only be able to stay in
orbit for about 23 days. After that, it will require a course correction to return its orbit to the plane. This is because point L1 is unstable, in that things placed at L1 will tend to move away from it. An object at L1 only has to move a tiny bit towards Earth or a tiny bit towards the Sun for it to begin falling toward the Earth or Sun.
Lagrange point L2, in the Earth-Sun system, lies on the line connecting the Sun and the Earth, close to Earth but just outside Earth's orbit. L3 is on the same line, on the opposite side of the Sun from the Earth, again just outside the path of Earth's orbit. At points L2 and L3, orbital inertia keeps objects in orbit. Since gravitation is pulling an object in orbit around L2 or L3 toward the two large masses, gravitational forces cannot balance the way they do at L1. L2 and L3 are unstable, since the plane that an object may orbit in around the Earth-Sun line will move as the Earth orbits the sun.
The other two Lagrange points, L4 and L5, can be found by
considering the equilateral triangle formed by the two large bodies, and a point in the orbital plane. If you are at Earth, and you face the sun, and then turn 60 degrees left or right, and travel a distance equal to the distance between the Earth and the Sun in the plane in which the Earth orbits the Sun, you will be at L4 or L5. L4 and L5 are stable points, as long as the mass ratio between the two large masses exceeds 24.96 (see footnote), which is not immediately obvious. This stability is the result of Coriolis forces, the same phenomenon which causes hurricanes. Normally, you would expect an object placed at L4 or L5 to wander off in a hurry; its not obvious that there's any reason for anything to stick around. As an object moves away from L4 or L5, it begins to rotate around the point, as a result of the rotation in the system.
Currently, NASA has a spacecraft at L1. SOHO is used to observe the Sun from an unrestricted vantage point. More about SOHO at is available from
Source for the 24.96 mass ratio:
http://map.gsfc.nasa.gov/html/lagrange.html, which also has some pictures of where the Lagrange points are, in case my explanations aren't enough.
For a mathematical treatment, refer this paper:
Even more Lagrange point information is available from the MAP Observatory site: http://map.gsfc.nasa.gov/m_mm/ob_techorbit1.html