A Jewish "priest" is called a Kohen in Hebrew (Kohanim in the plural). The original Priests were Aaron and his sons and all the descendants over the generations can trace their roots back to the original Kohanim. The "Kohanic gene" is transferred only through the male which means that a person is Kohen if his father was a Kohen etc. The Kohanim worked in the Temple and they were in charge of all the sacrifices and the day-to-day spiritual life of the Jews. Today, being a Kohen is simply a genetic issue and they are not necessarily spiritual role models.
A Rabbi, on the other hand, has taken exams in various aspects of Jewish Law and is supposed to be a spiritual role model to others.
The Torah in Numbers 27:8 relates that God told Moshe Rabbainu to confer on Joshuah the leadership. God told Moshe to place one hand on Joshua's head. However, Moshe placed both his hands on Joshuah's head as a sign that he was transferring the leadership with a full heart.
Smicha, ordination, was conferred during Talmudic times, and in order for a person to receive the title Rabbi and to sit on the Sanherdrin, a short ceremony was held where the semicha was given.
Today, when a rabbi receives his ordination he receives a diploma to prove that he has studied the required texts and passed exams on them. There is no specific time for studying the Texts, each person needs to learn at his own speed. It is that ordination that allows a person to become the Rabbi of a community and to earn a living in the field that he has invested his energies in.
The role of the Rabbi is to serve as the spiritual guide, leader and teacher of the community. He is required to be able to answer questions of a religious and a personal nature and to officiate at all Jewish ceremonies.