A random Vector field|field is a generalization of a stochastic process such that the underlying parameter need no longer be a simple real space|real or integer valued time, but can instead take values that are multidimensional vector space|vectors, or points on some manifold.
At its most basic, discrete case, a random field is a list of random numbers whose indices are mapped onto a space (of n dimensions). When used in the natural sciences, values in a random field are often spatially correlated in one way or another. In its most basic form this might mean that adjacent values (i.e. values with adjacent indices) do not differ as much as values that are further apart. This is an example of a covariance structure, many different types of which may be modeled in a random field. More generally, the values might be defined over a continuous domain, and the random field might be thought of as a function valued random variable.