Let's start with the entrance examination for a conservatory
For this kind of audition, you will normally have to play scales, one or two contrasting solo pieces and etudes. Some schools also ask for orchestral passages.
This is your chance to demonstrate your potential. Teachers will choose students depending on whether they think they can do something with you or not and how many places they have available in the class. It's not about being perfect and the most advanced students aren't always the ones getting the spots.
Teachers are looking for students who are not only talented but also confident, smart, well organized, hard working, dedicated and serious. You need to give the panel the impression that you have these qualities when you present yourself. They must not only like your playing, they must also like you. Play musically and take some risks, if something doesn't go well, ask if you can play it again, they want to hear what you can do so they'll give you another chance most of the time. Dress in a smart casual way without overdoing it! Answer all questions confidently without sounding arrogant. In other words, be polite and be yourself. That way, you should leave a good impression and maximize your chances to be accepted in the school.
It's a good idea to take a lesson or two with the teacher before the audition so you can see if you think you can learn something from him/her. Also, if the teacher already knows you and likes you, this will increase your chances.
After that,If you don't like the teacher, you better not waste your time auditioning for him/her!