Formerly the ruling military class of the Covenant, the Elites are proud, storied fighters with a long and interesting history. Their recent split from the Covenant is based on differences in religious interpretation of the "Great Journey" but the schism runs deeper than that. The Elites have allied with their former Human foes in part because of a deep-seated resentment of the Prophet rule and in part because they actually understand the scale of the Flood problem.
NOTE: Within the context of Halo 3, we see the Arbiter come to a realization about the supposed "true nature" of the Great Journey, but he does not openly communicate this to other Elites. 343 Guilty Spark begins to tell the story at the end of the game. Combined with this text, it would appear that the bulk of the Elites now believe that either the Great Journey does not exist, or that it is not what the Prophets have said it is.
That there is a separate and deep-seated resentment of Prophet rule does support one of my own long-held theories, which is that the Prophets intentionally manipulated the Elites into believing they shared the same faith in order to win a respite in hostilities that they could not decisively win by force of arms. In that context, the overthrow of the Elites by others more firmly under the Prophets' control, in this case the Brutes, was not only inevitable but was likely planned from the very start.