A field theory is a mathematical description of physical phenomena which describes the behavior of a particle, not only by the physical qualities of the particle itself (e.g., mass, charge, etc.) but also the position of that particle in space relative to other objects. Such descriptions are called "field theories" because the effects of a particle's position exist regardless of the presence of an actual particle, and so describe physical phenomenon at every point in space - such a description is a "field."
A "field theory" describes a force on a particle in terms of its position, or location in a "field". An example of a field theory is James Maxwell's theory of electromagnetism. Subsequent field theories are the theory of relativity, and modern quantum field theory.
Isaac Newton's (classical) theory of gravity is also considered a field theory, but it is based on "action at a distance" and does not have a force field. To the extent that a field theory has come to imply the existence of a force field, Newtonian gravity would not be one.
Newton described the force on a particle as related to the product of the particles mass and its distance from another mass:
This is traditionally called a field theory because regardless of the first particle, the second gives rise to a "field" everywhere in space which would affect any particle at any position.