One common source of anxiety for graduate students arises from not knowing how to address the faculty within their graduate program. What is polite and respectful? But not obsequious or sycophantic? It’s a tough call, particularly if you are new to your graduate program. How individual faculty members prefer to be addressed depends upon two variables: institutional culture and the faculty member’s personal preference.
I have one friend who works in a department that has a policy stating that all faculty members must require students to address them as either "Professor" or "Doctor," but that particular department doesn’t have a graduate program. My sense, though, is that such prescriptions are rare, especially at the graduate level. What is far more common is for faculty within a department to have individual preferences, which makes figuring out proper forms of address a more quirky, difficult navigational issue. Oftentimes a department's faculty will informally follow the same preferences, but there are frequently outliers. These outliers are the primary source of anxiety for grad students. An outlier, someone, for example, who absolutely insists that even his most advanced graduate students address him as "Doctor," can be the quickest to snap, scold, or correct if you address him by any title other than his preferred one.
If you are unacquainted with a faculty member and his preferences, you almost never will go wrong by addressing him as "Professor LastName." Some faculty, however, find "Doctor LastName" too formal or pretentious, and "FirstName" too familiar, informal, and presumptuous. "Professor" is a safe, happy medium that you can generally rely upon, until or unless individuals indicate that they would prefer to be addressed in a different way. While some may express a different preference from "Professor LastName," they will be very unlikely to perceive you as disrespectful or obsequious for addressing them that way initially.
Just as in conversation, when writing to a professor with whom you are unacquainted, or barely acquainted, for the first time, you’re pretty safe with "Professor LastName." Once the professor replies, you can generally gauge the most appropriate future form of address based upon how your correspondent has signed his or her own e-mail reply. For better or worse, this cartoon  may be even more accurate than its author realizes. One would hope that professors would sign the e-mail in accord with how they would prefer to be addressed by you. If your correspondent relies upon an automatic signature though, even an e-mail reply may not shed much light on the best form of address for that person. Hang tight with "Professor LastName" in that situation until you figure things out for certain.
Incidentally, I personally feel that relying upon an automatic signature borders on the rude, precisely because it does not permit the correspondent to gauge how I perceive the nature of our relationship. Alternatively, using an automatic signature with full title and contact information, but then typing a more personal “signature” above it is quite common, and does not strike me as rude in the least bit. And with close friends/colleagues (there often isn’t a difference in academe) I use an automatic signature in only the most formal or impersonal of circumstances; say, in an e-mail that might get passed along to someone with whom I have a less familiar relationship. Not signing any name at all, I feel, is not only rude, but also dismissive, unless the matter at hand is quite casual and the correspondent quite familiar.
Hopefully it goes without saying that you should never address a faculty member as "Assistant Professor LastName," or "Associate Professor LastName," titles that you might see in an e-mail signature or on an office door. While the professoriate is generally divided into the three ranks of assistant professor, associate professor, and full professor (this last of which is written simply as “professor”), these rank-denoting qualifiers are never used conversationally. To use them would be exceedingly weird, and possibly would be perceived as rude, as highlighting someone’s institutional status in an inappropriate way.
In all situations, pay careful attention to compound or hyphenated last names. Professor Smith-Baker probably prefers to be addressed as "Professor Smith-Baker," and not "Professor Baker." The hyphen is in there for a reason, so take note of it. In some cultures, compound last names, i.e., "Smith Baker," are used, and while there is no hyphen, the expectation is that both names are to be used. And plenty of last names are difficult to pronounce, in which case you should ask a fellow traveler in-the-know the proper pronunciation. Or, politely ask the faculty member how his name is pronounced. People can be quite sensitive to mispronunciations, and hopefully they will show you the same respect regarding your own name.
Once you have successfully defended your dissertation, after that initial round of congratulations and frequently bandied "Doctors" (which your close friends will likely succeed in articulating with exceeding sarcasm), and after you've managed to convince your nonacademic friends that you can’t write them Oxycontin prescriptions with a doctorate of philosophy, and when you show up for your first job, you're going to need to decide how you prefer for your undergraduate and graduate students to address you. I say "undergraduate and graduate" because, if you teach both types of students, your preferences might be different in each case.
The decision is an intensely personal one. Many beginning assistant professors are advised to have students address them by a formal title, such as "Professor," so as to establish firm boundaries and reinforce the new faculty member's professional ethos. If you’re initially unsure how you want to be addressed by students, you can always begin with "Professor LastName" and relax from there, if you so choose. As in most situations, it is easier to relax formalities than to suddenly install them.
In my own case I have resisted this advice, and I personally prefer to be addressed by my first name by all students. The formalities of a title do reinforce institutional hierarchies and make it easier for new and/or young professors to command respect in the classroom. However, I am more interested in having students begin to see themselves as adults than in reinforcing my institutional status. I feel strongly that part of the social development that takes place in undergraduate education involves students beginning to perceive themselves as adults, and learning to communicate in an adult-to-adult fashion, rather than in a student-to-adult fashion.
This feeling applies even more strongly for me in the case of graduate students. I want the graduate students I interact with to begin seeing themselves as intellectual peers and colleagues, if they don’t already. Admittedly, no matter how formal or informal you are about your title, there is still an enormous power differential between a tenure-track assistant professor and undergraduate or graduate students. I feel though, in my own case, that I am most invested in helping students to see themselves as adults and peers, rather than as only students.
And, in order to reduce students’ anxiety about the issue, I state my preference explicitly on the first day of class. I appreciate that students are being polite when they address me as "Doctor" or "Professor" or "Sir," which is how students usually initially address me here in North Carolina. But I also explain to my students that once someone has expressed a preference — first name only in my case — the polite course of action is to abide by that preference.
Given how many more legitimate worries there are to navigate in an academic career, knowing how to address faculty shouldn’t be one of them. If unsure, err on the side of respect, and adjust accordingly.