If you are inexperienced as an expert witness, you should know the following for deposition:
Know the legal issues. Speak with your client about the legal issues, and get comfortable with the legal terminology for your case.
Only one person may speak at a time. Everything you say at a deposition is being transcribed. The court reporter cannot make a good record when people interrupt each other or talk at the same time.
Only answer the question that was asked. Ordinarily, after you have answered the specific question that you were asked, you will not want to volunteer any additional information. Even if opposing counsel sits silently and seems to expect you to say more, don't fall into that trap. If opposing counsel wants more information, he will ask a follow-up question. Get used to silence even if it makes you uncomfortable -- it is your job to answer questions, not to fill silence.
Think before you answer questions. Whether on direct examination or cross-examination, think before you speak. It is almost always better for there to be a short pause before you answer a question than for you to give a bad answer.
You will be asked "hypothetical questions," which may be unfair. Opposing counsel may ask you to assume facts, and explain how your opinion would change if those facts were true. Do not assume that the facts are true -- the attorney may be trying to confuse you, or make you doubt your findings.
It is okay to answer, "I don't know." You may be surprised at a deposition. I heard a toxicologist complain that he was asked how his findings would be affected on the basis of a source of contamination that he learned about for the first time during his deposition. He was flustered, and testified that the contamination "could have a substantial effect" on his conclusions. When he calmed down back at his office, and analyzed the new information, he found that the effect was in fact minimal. But it was too late for him to fix his testimony and his client lost confidence in his ability to testify. The case settled for less than it was worth and he had to fight his client to get paid. Had he answered, "I don't know," or, "The analysis is complex, and I will have to perform a new set of calculations before I can answer that question," he would have avoided a costly mistake.
Do not overstate your client's position. An overstatement can devastate your credibility.
Pay close attention to questions. If you don't understand a question, ask for it to be repeated. You cannot help your client by answering a question that you do not understand.
You were not hired to impress people. You were hired to persuade people. It may be possible to do both, but if you must choose between looking impressive and being likable, it is better to be liked. You should think of yourself as an personable teacher, not as the ultimate arbiter of fact.
Be Careful With "Terms of Art". Most professions use common words to mean something other than what a layperson would understand the word to be. Make sure you understand the legal terms of art the lawyers and judge in your case may use, and make sure that the client law firm understands the terms of art that you may be using as you explain your case. Be alert for possible confusion, and be prepared to explain any terms of art that you use during testimony rather than creating a possible ambiguity.
Try to make your testimony understandable, and avoid technical jargon, but do not be imprecise. If you are testifying about an "adhesive" failure, you will not ordinarily want to refer to the adhesive as "glue" without first making a clear record of the differences between an "adhesive" and "glue," and explaining that you are using the wrong term to make your testimony more understandable. Otherwise, you can expect opposing counsel to use your testimony against you, to argue that you don't know the difference between "glue" and an "adhesive."
Don't follow opposing counsel's lead, if he is leading you astray. Staying with the "adhesive failure" example, opposing counsel may repeatedly refer to the adhesive as "glue," in the hope that you will follow his lead. The short-term goal is to trick you into believing he is unsophisticated about the scientific issues, and the long-term goal is to be able to argue that you don't know the difference between "glue" and an "adhesive." Make a record, early in the questioning, that the substance at issue was an "adhesive," and that you are assuming that the references to "glue" are in fact references to the adhesive.
Listen for "red flag" questions. If opposing counsel asks you a question that sounds like "legalese," it may be an effort to trick you into making a damaging concession. If you know the legal issues and standards governing the litigation, you should be able to avoid this type of trap.
When it is possible to be definite, be definite. If you always start an answer with "I believe" or "In my opinion," the effect may be to suggest that your opinions are imprecise and that it is perfectly reasonable for people to disagree with you.
Be careful about accepting the opponent's facts. During a deposition, opposing counsel is likely to ask you to agree with certain key facts. You should discuss which facts are "certain" with your client before the deposition, so that neither of you are surprised. Make sure you understand which facts are in dispute, so you don't accidentally concede a point which damages your client's case.