In short, he is the person who calls other men to the Torah during a Torah reading service (Shabbat morning, Shabbat mincha, Monday morning, and Thursday morning. He is essentially the "director" of the Torah service. He tells people when to come to the bimah (the stand from which the Torah is read), and generally conducts the people present at the bimah. He recites the prayers that are not said by those called to the Torah, such as the misheberachs (usually prayers for healing, but maybe also prayers for a person's birthday, anniversary, or upcoming wedding, etc) that are said between certain aliyot. (I've seen the misheberachs between the 3th and 4th aliyot, but also between two later aliyot.)
In addition to calling people to the Torah and leading some davening, he is a resource for the person actually leyning (reading) the Torah. He will have a book or print-out (usually a special book made for gabbais/gabbaim or leyning) that has both the vowels and the tune marks for the Torah reading. Remember that the Hebrew on the Torah scroll has no written vowels, so the reader must memorize the correct pronunciation of the parsha. Leying takes a lot of preparation, so appreciate that small group of people who regularly leyn for your benefit! Likewise, the Torah is chanted to a particular "tune" for lack of a better word. The leyning books have marks that could be analogized to musical notes above the letters.
So how does the gabbai use these books to help the person leyning? The gabbai follows along with the reading and corrects the Torah reader when he needs it or asks for the correct pronunciation. Likewise, the gabbai will correct the chanting itself. In some communities (or among certain individuals-like a father and his son being bar mitzvahed), hand signals can be used to "conduct" the chanting just like at the symphony. From my perspective in the "audience," I really enjoy watching the hand signals, and it helped me to get a better understanding of the chant by using two of my senses instead of only one (in other words, seeing and hearing).
Historically, gabbais have performed many other tasks, such as being a personal assistant to the rabbi or acting like a caretaker for the synagogue and grounds. In some communities, gabbais may still serve these functions, especially acting as an assistant to the rabbi.
Why am I bothering to tell you about what a gabbai is and does? In short, you should know who this is in your community. He can be a great resource for you (assuming he wants to be), and if you are ever lost during a service or in the synagogue, this is a person who can tell you when and where to go. The community's regular gabbai will generally be on the ball, as well as being knowledgeable both about Judaism and your particular community's customs. This is probably also a good person to consider approaching to ask for tutoring if your conversion requires a tutor (you should most likely offer some kind of compensation, of course!).