A physical field, i.e., what is termed a field within physics, is an entity consisting of some physical property of the space within some particular volume. A common use of the term is regarding the effects of gravity, i.e., a gravitational field (e.g., Earth's gravitational field), with other types being electric fields and magnetic fields. In each case, the physical property is some force that affects some objects within the field, and the effects of other forces (strong force and weak force) also constitute such fields. The natural way to describe such fields is using a corresponding mathematical field.
Generally such fields do not actually have a border, but fade to insignificance with distance from their strongest region. For example, Earth's gravitational field contributes to gravity at all distances, but its strength falls, and at some distance from Earth, the Sun's gravitational field is sufficiently strong that Earth's has very little effect. A gravitational field that is the sum of that of Earth and the Sun (and the other solar system planets) can be considered, as well as throughout the Milky Way and even the universe.
Objects can be thought of as "causing" fields, but also can be modeled as being fields, the object defined by its effect on its surroundings. Quantum electrodynamics (QED) can be thought of as a quantum theory devised to handle fields (such as electric fields), more-or-less by treating particles as fields and providing the means to describe the interactions of such fields (in such a way to include quantum effects).