What do interviewers look for in a candidate?

The most important thing to remember is that your interviewer will most likely also be a supervisor/tutor. Therefore, the best thing to do is to come across as someone they'd enjoy supervising and as someone who would benefit from the unique system of supervisions/tutorials in Cambridge and Oxford. This of course implies having to be polite and friendly, as with any interview, but also requires a few more specific qualities in a candidate. Firstly, and arguably most crucially, you should become used to vocalising your thought process. This is because the interview will likely manifest as a series of linked problems with increasing difficulty, and the interviewer will be looking to assess your thought processes along the way, while also guiding you if necessary. By vocalising your train of thought, you allow the opportunity for dialogue and, if need be correction, rather than inexplicably ending up at an answer, whether it be correct or incorrect. I would strongly recommend practising this prior to the interview, with a teacher or even alone, because it doesn't come naturally to most people, but will be massively beneficial in the interview. Secondly, your interviewer will be incredibly passionate about their subject and will be very excited to see the same passion in you, as the candidate. To aid this, try to demonstrate your enthusiasm for your chosen subject, both verbally, but also by preparing physical proof of your extra-curricular work. Perhaps write a brief project/EPQ on a relevant topic, or an extended essay. This will provide a unique point of discussion in your interview and will make you stand out in the memory of your interviewers when they review the candidates, while also demonstrating a clearly motivated desire to further your study of your subject. These two points, along with a sound understanding of your A-Level subjects, should put you in very good stead for your Oxbridge interviews.