As previous answers have said, fields are not algebraic. They are also not essentially algebraic, because categories of models of essentially algebraic theories have an initial object, and the category of fields instead has a set of initial objects -- Z and Z_p for each prime p. (There is no map from a field of one characteristic to a field of a different characteristic, so there can't be a single initial object.)
Fields are models of a a theory which is essentially algebraic plus allows specification of disjunctions. In the language of sketches, fields are the models of a "finite sum sketch." This was proved in Diers' thesis and is spelled out in the paper "The formal description of data types using sketches" by Charles Wells and Michael Barr, in volume 298 of the Springer Lecture Notes in Computer Science, 1988. For a general overview of sketches, see "Sketches: Outline with References" at http://www.cwru.edu/artsci/math/wells/pub/pdf/sketch.pdf .
ADDED 17 November 2009. The category of models of a finite-sum theory is not as nice as the models of an algebraic theory or even an essentially algebraic theory. Generally, the more different kinds of things you can specify in a sketch, the more awkward the category of models is. The category of fields is pretty awkward!
It does have filtered colimits and is closed under ultraproducts. Field theorists have made considerable use of the closure under ultraproducts.