Least action principle
The Lagrangian formalism is a universal approach to describe the evolution of a physical system. The basic idea is the least action principle, the idea that the evolution minimizes some quantity (for example the energy).
The resulting equations are called the Euler-Lagrange equations. This formalism can be applied to all known physical theories, such as classical mechanics, general relativity or quantum field theory.
A nice explanation in three steps (intuitive, example, abstract) can be found here: