Field

In PDE, a field refers to any function (or section) on the underlying spatial or spacetime domain. Common examples of fields include

Real scalar fields, which take values in the real line $\R$ . Examples include the field u appearing in NLW or KdV.
Complex scalar fields, which take values in the complex plane $\mathbb{C}$ . Examples include the field u appearing in NLS, or the field $φ$ appearing in MKG.
Vector fields, which take place in a vector space, or perhaps in some vector bundle over the manifold (such as the tangent manifold).
Connections, which look superficially similar to vector fields but transform differently under gauge transformations. Examples include the field $A α$ appearing in MKG or YM.
Tensor fields, which are sections of some tensor product of a base vector bundle. Examples include the metric g appearing in the Einstein equations
Spinor fields, which are sections of a spinor bundle, and which appear primarily in Dirac equations.

This notion of a field arises from physics and is unrelated to the algebraic concept of a field (i.e. a commutative division ring).