The process directors go through to choose their actors changes greatly from one director to another. Some directors look for the most skilled people, some directors look for people who fit what they have in mind for the character, some directors might use some other more personal and subjective means of determining, and some directors use some mixture of the above. Because the directors all use such different means to choose their cast, you pretty much just have to do the best you can and hope it's what the director wanted. You can certainly do things to improve your chances of being picked, by making sure your audition piece is refined and by choosing a piece that fits the character you're auditioning for. Ultimately though it's out of your hands and you can't feel bad if you're not what the director was looking for.
Usually you'll fill in an audition sheet that they'll give you before you audition. You'll be asked to give contact information on that sheet and they'll contact you based on that information.
Not everyone goes through callbacks, though you have to make sure you're available to go through callbacks if they ask you too. Sometimes a director will already know where they want to cast you (or that they don't want to cast you) and won't require you at callbacks. Callbacks is where they do more pointed exercises and/or cold reads to help them finalize any parts they weren't sure about after the initial auditions.
Hope that helps,