The Rabbi leads the order of the service and the Cantor leads the prayers.
The cantor.
A Cantor Sings in the worship services in a Synagogue. It's a Jewish person who worships and leads the singing in a synagogue. I am actually reading a book about it right now. Very interesting ;)
The guy who leads services at synagogue.
Technically, any congregant who is knowledgeable may lead the prayers. On Shabbat and festivals, there is often a cantor (Hazan) who leads the prayers.
According to some customs, that persons stands at the central bima (table), where the Torah is also read. That is the practice I've seen in Sephardic synagogues. In another custom, there is a lectern adjacent to the Holy Ark in the front of the synagogue, and that is where the leader of the prayers stands.
Any Jewish person over the age of 13 can lead Jewish services. In orthodox tradition, this person must be a man. Rabbis are Jewish teachers, and in most synagogues the Rabbi leads the service, though this is not a requirement.
No one leads Channukah. This is a holiday that's celebrated in the home; and certain prayers are added to the regular synagogue services too.
Any Jewish person over the age of 13 can lead a synagogue service, as long as at least 10 Jews are present. Often the Rabbi or Cantor leads the service, but this is not a requirement. In orthodox tradition, only a man can lead a service, unless there are only women present.
The Yom Kippur morning services.
The equivalent word is "synagogue service."
The hazzan leads the prayers in the synagogue and often is the one who also reads from the Torah-scroll.
If you are a guest at a bar mitzvah, you don't really do anything except sit with the congregation during the service. The bar mitzvah boy (or bat mitzvah girl) usually leads and/or takes part in the service, reading from the Torah or at least saying the Torah blessings.