Lagrange points are found in all gravitational systems
with two large bodies and one small one. The important case for space colonization
is the Earth-Moon system
, with the space habitat
as the third body. The Italian-French mathematician Joseph Louis Lagrange
discovered the existence of these points in his pioneering 18th-century work in celestial mechanics
There are five Lagrange points in each three-body system. The interesting property of these points is not that the gravitational force cancels out (which only happens at one point) but that a small body placed there will stay locked in the same position relative to the two larger bodies, without using any fuel to maintain the orbit.
Well, not exactly--only two of the five points are stable. If you nudge the small body away from a stable Lagrange point, it will drift back. The other three are unstable: if you nudge the small body, it will drift away. It's the same as the stable resting place at the bottom of a pit and the unstable resting place at the top of a hill.
You can find the two stable points by drawing a line between the large bodies (say, Earth and Moon) as the base of two equilateral triangles above and below the line. The points at the tips of the triangles are the two stable Lagrange points.