The question and answer period after your talk is just as important as the talk itself. If you have given a good talk—a talk that engages most members of the audience—then you should expect to get a lot of questions. In fact, I think this is one of the clearest signals that a talk has succeeded. Nonetheless, the question and answer period can be terrifying, especially for less experienced speakers. "What if someone asks a question I can't answer?" "What if someone disagrees with what I have said?" "What if someone is downright mean or is trying to show off their superior knowledge?" All of these scenarios happen, so it is good to be prepared. I can't anticipate every scenario, but I can offer some general advice for the major categories of questions.
First, always repeat the question, both to ensure you understand the question and so that everyone in the room is participating. Remember, this is not just a conversation between you and the questioner, it is a discussion for everyone in the room. Feel free to rephrase the question with plain language, either to make it easier to understand or—if you feel the question will take the discussion too far afield—to turn the original question into one that you would like to answer! This is still your talk. So take the prerogative to make the answers relevant to your talk. If you can't answer a question (for any reason: you don't know, it isn't known, it isn't possible to discover), then be honest and say that you don't know and why, but always feel free to turn the question into a related one that you feel you can answer with confidence.
Keep the tone and scope of your answer similar to the tone and scope of your talk. Answer with plain language to the extent possible. Don't use the question and answer period as a time to get into the nitty-gritty technical details. If the question absolutely requires a technical answer then tell the questioner to meet with you later to discuss these details. Say something like “That’s a great question, but it will take me a while to explain all the technical details. I would really like to hear your feedback on what I have done, so let’s meet after the talk to discuss it further.” And then, if possible, give a more general answer in plain language that addresses the topic of the question. Of course there may be some forums where everyone in the room wants a long technical answer, but in that case, you probably will know the audience well enough to know that that is what they want.
Sometimes people ask simple questions that make it obvious that they missed something you said in the talk. They may have zoned out or you may have explained it poorly. In any case be polite and answer the question as if they had not missed it.
If presented with a question that completely misses the point of your talk, then this may not be entirely their fault. Probably partly you are to blame. So again try to be polite and then feel free to extend their question to a realm that will interest those who were paying attention more closely. Then answer the new question you have posed.
Always remember that the question and answer period is part of your talk. So don't let it be hijacked by someone asking a question persistently. If a questioner repeats their question after you answer and you truly did not answer that question (it happens), then try to answer again. If they come in a third time, then tell them you will follow up after the question and answer period to clarify. Sometimes, questioners and presenters seem to talk at cross purposes during the question and answer period and the conflict can be resolved only after a long one-on-one discussion. The earlier you can spot this potential scenario, the more useful it is to divert the discussion to later.
If you are nervous in the question and answer period, then you are in good company. Many people do not have their strongest ideas when they are standing in front of a group of strangers. I can’t tell you how many times I have thought of a better answer to a question two hours after the talk ended. The most important thing, therefore, is to be open to the fact that your audience has a diverse and different intellectual background than you do. The question and answer period is a time for you to learn from them. If you feel a question is challenging your work in a pointed way, but you can’t think of a good rebuttal, then don’t get defensive. Instead say something like “That is a really interesting point, I need to think about that some more. Can we discuss this further offline?” Then, think about what they said and follow up. You will likely learn something valuable.
There are many other kinds of questions that, for a variety of reasons, you may not want to answer at that moment. For example, sometimes people ask questions that are not really related to your work, but are related to some other talk they heard recently and to which you were not privy. Again, if you don’t know the answer, say so. Or, a questioner may ask you to extrapolate from your specific data to a much larger general problem. This is one of my pet peeves. When I give a talk, I extrapolate as much as I am comfortable extrapolating. However, some scientists like to extrapolate much further than I do and they may ask you to do this in the Q&A. When you answer these questions, make it absolutely clear that the answers are not well supported by the data you have collected and that they are largely “hand waving” arguments that require a lot more work. This is your “get-out-of-jail-free” card. In my opinion, it is more acceptable to extrapolate in this way when asked directly in the Q&A than to litter your main presentation with unfounded conclusions.
Sometimes, people ask pointed questions to explore your current investigations or future plans. This happens a lot during job talks. It may also happen if the questioner is a competitor (known or unknown). If you are happy to tell everyone everything about your work, then this question is not a problem. Personally, that is my approach to answering this question. I have not yet been “burned” or “scooped” by taking this approach to science. On the other hand, I study pretty esoteric problems. I know that there are many scientists, especially young scientists, who fear that by revealing their current work they may make themselves vulnerable to being scooped by a large laboratory. Answering this question requires a delicate balance of confidence and confidentiality. I think it is fine to point out that you are still working on a problem, without revealing specific details. You do not want to sound defensive. So, I would not say “I don’t want to talk about this because this is an extremely competitive field and I fear being scooped.” Instead, you might say something like “I agree that this is a really interesting problem and I am continuing to work on it. However, I can’t tell you much more at the moment because all of my new data are extremely preliminary and I am not confident that I can draw robust conclusions from what I have observed so far. I look forward to talking about this new work in the future.” Then, perhaps you can pull out the old trick of re-phrasing the question as something you would like to answer, and then answer your own question!