Find a proper space for the audition. If you are working with a casting director, he probably will secure a good location for you. When you work on your own, however, look for a space that is large, uncluttered and does not have a lot of background noise. A soundstage is ideal but not necessary. Also be sure to clear the area that will be used of all unnecessary furniture and debris. A nondescript background will help keep you focused on the performers.
Obtain or purchase the necessary recording equipment. If you are just auditioning vocalists, a DAT or other digital audio recorder with a good microphone will be sufficient. However, if you are auditioning actors or performers with some visual element to their work, use a video camera. For video, auditions do not require anything too fancy such as a RED camera. A handheld camcorder mounted to a tripod should be sufficient. The built-in microphone on a video camera may be adequate in a smaller space or if you aren't too concerned about sound quality. Otherwise, consider augmenting the camera audio with a more professional device such as a Lavalier microphone that can be attached to a performer's lapel. For auditions with multiple participants, consider using a boom microphone (as well a boom operator).
Obtain recording media. For audio, that may mean obtaining DAT tapes or simply outfitting your laptop for sound recording. Depending on your video camera, you may need to purchase the applicable media or it may already have a built-in drive for storing images.
Put an ID on the recording to identify the performer before he auditions. For video recordings, use a clapboard with the date, performers' name, production title, the specific role and any other pertinent information for your project. For audio recording, you can just speak all of that information into the microphone before proceeding. After the audition is over, be sure to label all tapes and other media with dates and other relevant information.
Consider using an online audition delivery system (if you have the budget). With a software package such as "Now Casting," you can post audition recordings online so that you (or any other members of your production team) can listen and watch auditions on the web. This can be an especially effective comparison tool as well as pointing out strengths and weaknesses of specific individuals.