On audition day: Warm up your voice AND your body before you go to the audition with a good 20-30 minute vocal warmup and several overall body stretches. This will help calm your nerves and allow you to perform at your best. Be sure to eat a light meal for energy - not right before you sing, but 2-3 hours prior to the audition. Steer clear of dairy and carbonated drinks, though.
In the audition waiting room: Don't engage in small talk with other auditioners in the waiting area. Be cordial and certainly not unfriendly, but you need to stay focused, and you can get sabotaged by those you meet in the waiting room. Some of these people are likely going to be your competition, so don't give them the edge by allowing them to drain your mental energy or derail your confidence just before you walk in.
In the audition room: When you walk into the audition room, your greeting to the directors should be a brief "hello" that is cheerful, energized, confident, and relaxed. Don't babble on or make excuses about being sick, etc... They are already judging the type of cast member you might be, based on that greeting. Don't give them any reason to think that you are unsure of yourself, or worse, a whiner or complainer. They want you to succeed, but they are looking for people who can do the job with the least amount of hand-holding...NOT someone who is potentially"high maintenance".
Before you leave: This is one tip that so many overlook. Make sure you find out before you leave the audition (ask the monitor, if you aren't sure) the names of the people for whom you sang. If it's an Equity audition, it should be listed on the audition notice, but sometimes an additional casting person is attending who may not be listed. Be sure to ask so that you can follow up after the audition with a hand-written thank-you note to each of person for whom you auditioned. Yes, it's old-fashioned, but it shows you have respect for them and the time and consideration they gave you. Whether or not you get the part, they will remember you for the thoughtfulness, and chances are, eventfully, if you continue in this business, you will stand in front of these same people again. Directors and casting agents see so many people, so anything you can do to make yourself stand out (in a good way) is worth doing. You may have to do some investigating to find a mailing address for them. If you can't find a physical address, email, while not as personal, is an acceptable alternative.
Just before you go in: A few moments before you go into the audition room, stand up, walk around, and even jump up and down in place several times - just enough to help you loosen up, get your breath flowing and blood moving. You never want to sit waiting for an hour or more at those dreaded open calls, then stand up and immediately walk in and sing. Also, you can do lip trills just before entering the audition room. This will get your low, diaphragmatic breath control engaged, ensuring that your first note will be "on the breath".
Post audition celebration: Finally, have something fun planned for yourself to do just after the audition. This is especially important for kids in the biz, but also applies to adults. Let's face it, most performers are hyper-critical of themselves. We are perfectionists, and we obsess about every little aspect of the audition after the fact. "Why did I wear that stupid dress?" "I should have sung the other song I had prepared." And on, and on.... These thoughts play over and over in your head like a broken record immediately after a big audition. If, however, you plan something fun to do post-audition, i.e., lunch with a friend, go to see a good movie, even simple retail therapy - you may not completely stop obsessing over your auditions, but at least changing your focus can help you deal with the nerve-wracking aftermath. :-)