A field equation is an equation in a physical theory that describes how a fundamental force (or a combination of such forces) interacts with matter. The four fundamental forces are the gravitational force, the electromagnetic force, the strong nuclear force and the weak nuclear force.
Before the theory of quantum mechanics was fully developed, there were two known field theories, namely gravitation and electromagnetism (these two are sometimes referred to as classical field theories, as they were formulated before the advent of quantum mechanics, and hence do not take into account quantum phenomena).
The first field theory was Newton's theory of gravitation, which described gravity as obeying an inverse square law. This was very useful in describing the motion of planets around the Sun. This theory is now superceded by Einstein's theory of general relativity. The field equations of gravitation are Einstein's field equations which describe gravity as a 'curvature' of spacetime produced by the existence of matter.
The electromagnetic force is best described by Maxwell's theory of electromagnetism. The field equations of classical electromagnetism are Maxwell's equations which describe how electromagnetic fields are produced from charged particles.
Modern field equations tend to be tensor equations.