A Lagrangian point (or Lagrange point, L-point, libration point, or Lagrangian libration point) is point in the vicinity of two objects orbiting each other at which a less-massive third object would not be drawn toward either of the two more-massive orbiting objects, i.e., the three would orbit stably, keeping their relative distances. Natural objects sometimes remain stably in orbit at such points, and they are used as places to locate spacecraft.
An obvious such point is between the two more-massive bodies where their gravity balances any acceleration due to its not being centered between them. This point, known as L1, is unstable: any slight movement from the point toward either move-massive object puts the less-massive object in the influence of forces that will take it toward one of the two more-massive objects. Other points:
L4 and L5 can be 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. The condition for their stability (Lagrange point stability) is a specific minimal mass ratio, the minimum being a number close to 25.
L1, L2, and L3 are unstable, but the positive feedback away from the point acts slowly, and can be counteracted by an occasional small push, a strategy that spacecraft use to park in these spots: a reasonable amount of propellant (light enough to be launched and carried there) can keep a spacecraft at the point (stationkeeping) for years or a decade.
The stable points naturally support orbits encircling or crossing the point. The unstable points also support such orbits (e.g., halo orbits) given the occasional propulsion to counteract the instability.
L4 and L5 points, which are also known as Trojan points, can collect debris since anything drifting to them 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. A Trojan asteroid is one orbiting the Sun in such a position (by default, assuming an L4/L5 relation with Jupiter). The term Trojan planet refers to an extra-solar planet residing in another's L4 or L5 point (as of 2020, no candidates have been widely accepted).
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: