In quantum field theory, fermions are described by anticommuting spinor fields. A four-fermion interaction describes a local interaction between four fermionic fields at a point. Local here means that it all happens at the same spacetime point. This might be an effective field theory or it might be fundamental.
Some examples are the following:
A nonrelativistic example is the BCS theory at large length scales with the phonons integrated out so that the force between two dressed electrons is approximated by a contact term.
In four space-time dimensions, such theories are not renormalisable.