A Lagrangian point (or Lagrange point, L-point, libration point, or Lagrangian libration point) is point close to two objects orbiting each other where a small third object would not be drawn toward either of the two larger orbiting objects. An obvious one is between the two where gravity between them and inertia of the inward acceleration equalize the forces toward each of the two larger objects. This one, known as L1, is unstable: any slight motion away from the point puts the small object in the influence of forces that will take it toward one of the two larger objects. Other points:
L4 and L5 are stable in that any slight motion causes forces to bring them back to the L4 or L5 point respectively. For example, an object in the L5 point for the Earth and Moon, if it drifted toward the Moon, the Moon's gravity would draw harder, but that would speed its orbit around the Earth, causing its Earth orbit to expand, which in turn speeds its orbit around the Moon, causing its lunar orbit to expand, bringing it back to L5. L1, L2, and L3 are unstable.
L4 and L5 points, which are also known as Trojan points, can collect debris since anything drifting close will tend to stay. In the solar system, some moons and asteroids are in such Lagrange points. I imagine the name libration points is used because and an object can sit in a small orbit around any of the stable points, thus oscillating.
All the points for the Earth, Sun, and Moon are candidates for space observatories and other satellites. Sun-Earth L1 is popular for observatories watching the Sun and Sun-Earth L2 is popular for observatories that are not. Some examples of current and planned space platforms: