This answer is based on my experience only, which is primarily with Orthodox synagogues.
Many synagogues have many members who happen to have completed the requisite course of study and become rabbis, but are not employed by the synagogue and work as, for example, accountants or bootblacks. But you don't seem to be asking about such people, so the rest of my answer will be devoted to rabbis employed by the synagogue.
Most synagogues have but one rabbi. He advises the synagogue board, and in many cases makes decisions binding on the synagogue, regarding spiritual matters. He also answers members' (and others') questions about all matters religious or pastoral, officiates at members' (and sometimes others') lifecycle events when asked (though most are not trained to circumcise), gives lectures in the synagogue, and sometimes leads synagogue services.
Many synagogues have two rabbis. The second fulfills the above roles when the first is unable; this is common in large congregations where there may be many such tasks to do. Sometimes the second has a designated role as, for example, youth rabbi, in charge of youth programming and lectures and leading youth-oriented services; or as, for example, programming director. Other times his title is simple "assistant rabbi" or the like. (Many synagogues hire programming directors or the like who are not rabbis, too.)
And some synagogues have more than two, especially if the congregation's size supports and requires it. And some have none at all.
Some synagogues have a rabbi listed as "emeritus" or sometimes as "senior". He does little or no work and is typically the former rabbi of the synagogue, who still draws a stipend for his past work. But "senior rabbi" can also mean the main rabbi of a congregation.
Also, occasionally, a rabbi-in-training serves a temporary stint with a congregation.