Modulo a prime ideal you're only guaranteed an integral domain. The residue ring is only a field when the ideal you're taking as a modulus is maximal. All maximal ideals are prime, but the converse is not true - for example in the ring Z[x] of polynomials with integral coefficients the ideal (p) generated by a rational prime p is prime but not maximal.

There are of course many rings in which all prime ideals are maximal. The integers, for example, and more generally the integer ring of any number field (which are not always principal ideal domains, but are examples of Dedekind domains).