Such fields will contain infinitesimal and infinitely large elements, suitably defined.
We say that F satisfies the Archimedean property if, for every two positive elements x and y of F, there exists a natural number n such that nx > y.
In a non-Archimedean ordered field, we can find two positive elements x and y such that, for every natural number n, nx ≤ y.
Max Dehn used the Dehn field, an example of a non-Archimedean ordered field, to construct non-Euclidean geometries in which the parallel postulate fails to be true but nevertheless triangles have angles summing to π.
Sometimes the term "complete" is used to mean that the least upper bound property holds, i.e. for Dedekind-completeness.