Today’s advice is written specifically for undergraduates, even though graduate students and postdocs may benefit as well. (I certainly didn’t figure these things out until I was a faculty member myself.) What is the best way to approach professors by email, in particular if you want or need something from them? I’ll give you the most important advice up front: Be absolutely clear about your intentions. Tell me exactly what you want or need from me, and why. This will save both of us a lot of time, energy, and frustration. (Frustration would be mostly on your part, though, when I silently ignore your messages.)
As a professor, I am constantly being inundated with requests, propositions, recommendations, and demands. And most of them fall into the category of spam, or at least not very high on my priority list. Almost daily, I receive requests from people who would like to join my “esteemed laboratory.” They usually come from overseas, sport a long list of qualifications in the latest experimental techniques, and seem to have no knowledge of the fact that my lab is entirely computational (it says so right on our homepage). I’m also constantly asked if I want to participate in this or that activity, survey, or committee. (Do I want to be on the university committee that decides on the color of the men’s changing room in the new gym? No.) Over time, I’ve gotten quite good at deleting or at least ignoring these emails. Usually, I don’t need to read past the first or second sentence before I send the message to the circular filing cabinet. If you want to reach me, you have to somehow make it through that noise. Your message has to be clearly different.
Now, reaching me through this noise may seem a daunting task, but it’s actually not that bad. The truth is, I want to interact with you, I truly do. For example, I have over 10 people in my lab, from undergrads to senior scientists, and almost all of them got there by sending me an email at some point. They all got through to me. In order for me to be able to help you, though, to consider your request, I need you to help me. I need you to be crystal clear in what you need or want. Do you need to see me before Friday because you want to drop my class before the deadline? Say so. I’m not going to chew off your head. (An aside: Students seem to be extremely shy about requesting a drop. I really don’t care. If you want out of my class, I won’t be in your way.) Would you like to meet me because you’d like to work in my laboratory? Say so. If you’re an undergrad, say whether you are looking for a paid job or are willing to volunteer/work for course credit. If you’re a graduate student, tell me why you are interested in my lab and what topics you’d like to work on. In general, the more clearly you can spell out what it is you want from me, the quicker I can decide whether I can be of help or not. And if I receive a clear, well-formulated request that I have to deny, I’ll generally respond quickly with a “no,” and possibly with suggestions where else you could go. By contrast, I’ll generally just ignore or delete any messages whose intent is not clear to me.
If you would like to request a meeting with me, please suggest possible times that would work for you. Don’t just write “I would like to meet with you.” Write “I would like to meet with you. I’m available this week Wed. before noon or Thurs. after 2pm.” This saves us one round trip, because I can immediately suggest a time that might work for both of us.
Finally, if I ignore your message, don’t take it as a “no,” try again. I may simply have overlooked it. Or I may have had the best of intentions to respond to you, but then I got distracted by something, then I had to teach a class, and by the time I was back in my office your email was long forgotten. So if I don’t respond to your message, send me another one the next day, or maybe two days later. In general, I think you should try at least 2-3 times before you conclude that your message is being actively ignored.
Update: I wanted to add some thoughts about how to address professors in email. If you're an undergrad, I think you should write "Dear Dr." + last name. In my case, that would be "Dear Dr. Wilke." If you're a graduate student or postdoc in my department, first-name basis is fine, so "Dear Claus" or just "Claus." If you're a graduate student or postdoc at a different institution, I think the formal "Dear Dr. ..." is the better option, in particular if we've never had any interactions before. (If we hung out at a conference over a beer and talked until 2am about everybody and their dog, "Claus" is probably fine.) If you choose the formal address, pay attention to the response. If the response is signed just with a first name, you can assume you're on first-name basis from there on out.