  El Kaoutit and G\'omez Torrecillas introduced comatrix corings, generalizing
Sweedler's canonical coring, and proved a new version of the Faithfully Flat
Descent Theorem. They also introduced Galois corings, as corings isomorphic to
a comatrix coring. In this paper, we further investigate this theory. We prove
a new version of the Joyal-Tierney Descent Theorem, and generalize the Galois
Coring Structure Theorem. We associate a Morita context to a coring with a
fixed comodule, and relate it to Galois-type properties of the coring. An
affineness criterion is proved in the situation where the coring is
coseparable. Further properties of the Morita context are studied in the
situation where the coring is (co)Frobenius.
