To do well on an oral exam requires a rather different set of skills than those called for on a written exam. You still need mastery of the material, of course, but you need to access and articulate the material in a different way.
Most people find oral exams harder than written, but some really enjoy them!
In a well-designed written exam, the examiner has decided in advance what set of knowledge and skills is to be tested. Problems are designed to be completed within the given time limit, but some false starts and backtracking is expected. Each problem has a known correct answer.
The oral exam differs in each of these characteristics. Although the scope of the exam is given in advance, the examiner usually chooses questions dynamically, based on how the student answers the previous questions. Problems are designed to test the limits of the student's knowledge: often, an examiner will continue to ask questions in a particular area until a student no longer responds correctly. Examiners design problems (or give hints) that elicit the correct approach on the first try, without backtracking. And often the questions are open ended, asking for opinions or ideas for future work or aspects of the research area that the student has not yet considered.
Because of these factors, preparation for an oral exam requires a new approach, even for seasoned veterans of written exams.
You need to think on your feet. This is great if you have the skill. Otherwise, you can only compensate by great preparation.
Some previous experience is helpful: teaching, debating, oral presentations, tutoring, class participation, recitals, science fairs, etc.
The exam is usually tailored to the interests of the particular student. In this case, it is important to have a clear meeting of the minds between student and examiners on what material it included in the exam. Usually this is done by preparation of a written syllabus by the student, with the advice and consent of the examiners.
Sometimes the exam begins with a presentation by the student of initial research results or background material for a given research area. This talk should be carefully prepared, usually using slides. The presentation should be clear, well-rehearsed, and succinct. Your visual aids should be carefully organized and easy for you to navigate. Give some indication of what the problem is, why it is important, what you have accomplished, and what you hope to accomplish in the future. Sympathetic colleagues or your advisor can give very helpful advice if given a chance to listen to a rehearsal.
The rest of the exam is less structured. The examiners rotate, asking questions in turn. Sometimes one examiner will ask an entire set of questions in a row; other times the examiners interleave their questions with others.
Listen carefully to questions and make sure you understand exactly what is being asked. Follow instructions exactly - if a short answer is requested, keep it short. If more detail is desired, give a longer response.
Don't interrupt a questioner. Wait until he/she finishes the question before you start to answer.
Good examiners will ask you questions on a given topic until they tire of it or until you answer incorrectly. It is all right to be wrong--the purpose of the exam is to discover the limits of your knowledge, and it is all right to have finite limits! Students pass even if they don't know everything asked!
An important rule is to pause briefly after each question is asked. Take just a moment to compose your thoughts, decide what notation is necessary and appropriate, and organize your answer. If you do not understand the question, ask the questioner to rephrase it, or give your interpretation and ask if that is what is meant.
If you are sure you cannot answer the question, it is best to admit that and go on, rather than wasting time and focusing the examiner's attention on what you don't know rather than what you do.
If you are confident about a question, answer as directly as you can, but feel free to make comments about the relevance of this result to your work, etc.
Remember that each new question is a fresh start. Let the old one go. Don't get flustered--remember that the examiners expect you to be unable to answer some questions--that's how they explore the limits of your knowledge.
If you do find yourself losing your composure, ask the examiners for a brief break to get a drink of water or to sit down for a minute. You may be reluctant to delay them this way, but it saves time in the long run to get an accurate assessment of your abilities the first time.
Remember that an oral exam is an exhausting experience--comparable to running a marathon. Pass or fail, try to give yourself a break on other activities immediately before and after the exam.
If you believe that the question covers an area not on the syllabus, it is best to state that directly and non-belligerently, but then answer the question if you can. If it does make the difference between a pass and fail, then you are on record as objecting to the question before failing the exam, and this lends credibility to the objection.
Occasionally, you may find committee members more intent on impressing or belittling each other than they are on exploring the extent of your knowledge. There is not much you can do about this other than staying strictly neutral and trying to avoid assembling the same group for any subsequent exam.
Bias can also be a factor in your exam performance. If you believe that an examiner was predisposed to fail you, try to document how that person's examination of you differed from his/her examination of some other student. Consider trying to get the person excluded from your next exam.
If you fail the exam, make an honest assessment of your weak areas and any weaknesses in your presentation style. Practice. Study. Ask advice from your advisor. Try again.