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, 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. 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.