L1, L2, and L3
Both L1 and L2 are located on circumference of the planet's Hill Sphere. L1 lies on the line between the star and the planet, and L2 lies on the same line on the far side of the planet form the star.
At the L1 location, the pull of the star’s gravity is cancelled out by the gravitational pull of the planet from the other direction. The net effect is that an object orbiting at this location will behave as if it were orbiting a less massive star, and it will have an orbital period equal to that of the planet, rather than the shorter period one would expect from an object orbiting at a smaller average orbital distance.
At L2, the combined masses of the star and the planet act together, as if the object were orbiting a more massive star, so its orbit also has the same period as that of the planet, rather than the slower orbit one would expect.
The distance from the secondary mass to its L1 and L2 points, being based on the Hill Sphere of the secondary, grows with the distance of the planet from the star. (The farther the planet is from the star, the farther its gravitational field will be dominant before being overwhelmed by that of the star). If Earth were twice as far from the Sun, the Hill Sphere radius would be twice as far from Earth, at about 0.02 AU (see this blog entry for the equation).
The third point, L3, lies on the planet’s orbit, directly opposite the planet on the other side of the star. This would seem to indicate that any object orbiting the star at the L3 point would remain forever hidden from the view of observers on the opposite planet, and this has led many science fiction writers to hypothesize an alternate (or evil‑twin) Earth existing at the L3 point. Astronomers have been able to verify, however, that no significant object orbits at this location  (at least in our Solar System).
All three of these locations (L1, L2, and L3) are technically mathematical points, and as such are not stable orbital locations for any objects of significant size, unless such objects are able to artificially maintain their position. Objects are, however, able to follow very special orbits, called Lissajous, Lyapunov, or halo orbits, centered around these points .
Very simplistically, Lissajous orbits  are quasi-periodic—meaning that there is an element of unpredictability, such that the object cannot be accurately predicted to be at any given point in its orbit at any given time. (This means these orbits are non-Keplarian—not predictable by Kepler’s laws of planetary motion.) These Lissajous curved paths are highly inclined to the plane of the primaries.
Another kind of orbit which bodies can follow at L1, L2, and L3 are called Lyapunov orbits. These are periodic orbits that lie entirely in the plane of the primary bodies.
Finally, halo orbits are periodic orbits which are inclined to the plane of the primaries.
In our own Earth-Sun system, several satellites are currently stationed in orbits around the L1 and L2 points . Similarly, satellites could be placed in orbit near/around L3 on the other side of the Solar System. Such satellites would, for instance, provide the ability to monitor the surface of the Sun, and observe Solar phenomena 12 hours earlier than they would be visible from Earth.
This would allow early warning of events such as coronal mass ejections which would be able to affect the Earth. For such a system to work however, the L3 satellite would have to be able to “bounce” signals through a “repeater” satellite at either the L4 or L5 point (see the next section). Such a system isn’t technologially practicable for the Earth-Sun system at the present time, but a more advanced civilization may very well make use of such an arrangement.
L4 and L5
In our Solar System, in Jupiter's orbit, there is a large collection of asteroids at each of the L4 and L5 points. Those “ahead” of Jupiter are called the Trojans, and those “behind” Jupiter are called the Greeks. These actually occupy large, elongated curved regions, shaped somewhat like curved teardrops. Within this region, the asteroids actually librate around their respective equilibrium points (L4/L5 points), following what are sometimes called “tadpole orbits”.
It is also possible (depending on other gravitational influences) that two planets could share an orbit, each located at the other’s L4/L5 point. The long-term stability of such co-orbital objects would be highly dependent on other bodies in the system.
A planet can also have moons sharing an orbit at these same points. Saturn, in fact, has two sets of moons sharing orbits; Tethys has smaller moons—Telesto at its L4 and Calypso at its L5, and Dione has Helene at its L4 point and Polydeuces at its L5.
It is possible that Theia, the Mars-sized mass that impacted the early Earth, leading to the formation of the Moon, may have originally orbited at either Earth’s L4 or L5 point, but had its orbit disturbed, leading to the collision .
3. Pronounced "lisa-zhoo"