The race of the Sanghelli (Elites) joined the UNSC after the recommissioning of the Honor Guards (Bodyguards to the prophets) initiated, namely, by the Prophet of Truth, leading to their severance from the Covenant, and the genocide of their entire race. Their choice to join the UNSC was likely caused by the fact that they had been betrayed by the Covenant religion and sought vengance, lead by Thel Vadam (The current Arbiter in the 2nd and 3rd installments of the Halo trilogy), this became a reality. The rest is history.
The Elites never joined the UNSC. The Elites and other alien species that defected from the Covenant were known as the Seperatists and they sided with the UNSC after the second battle of New Mombasa. Also, the Elites didn't leave the Covenant because of the recommissioning of the Honor Guard; they were forced out because the Prophet of Truth saw the Jiralhanae (Brutes) as unquestioning, loyal pawns, rather than the intelligent, honorable Elites. A complete explaination of the entire situation is on the other end of this link. It is the Halo Nation page "Great Schism."