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An effective field theory (EFT) is a simplification of a theory of a physical field aimed at giving a usable approximation within a particular regime (e.g., within a specific spatial scale). EFT replaces some of the "hard parts" of the formulas with simpler expressions more amenable to manipulation and calculation. EFTs are used in basically any branch of physics that includes difficult field equations, including general relativity, particle physics, quantum mechanics, statistical mechanics, and fluid dynamics.
The above EFT description is loose and would fit the role of classical and non-relativistic physics's relation to more modern physics. However, the term was coined and is generally used for more modern approximation methods such as those developed to deal with regimes not amenable to classical physics, e.g., using an approximation of the difference between the classical and the "modern physics" result. Within certain areas of physics, particular EFT methods have been developed, and physicists in that area use the term EFT with a more specific meaning. One difficult problem tackled by EFT is many-body quantum field theory (QFT) and an example is the exotic matter presumed to be in the center of neutron stars. Chiral EFT is the name of a method used to tackle it.
Occasionally you see an assertion (claim) that some well-established model is merely an effective field theory. This is saying that the model merely approximates some as-yet undiscovered model that more-closely matches reality. Such statements have been made about general relativity, about the standard model of particle physics, and likely about every other well-known, well-accepted model.