
1. Don't neglect to practice. Whether you sing or play an instrument, practice is key to being relaxed. The more familiar you are with what you are performing, the less anxiety you will have about messing up.
2. Don't back up. If you mess up in the middle, or any place in your piece, don't back up and repeat the offending passage. Keep going. Chances are your audience didn't even notice.
3. Try not to be critical of your technical skills. Focus more on your overall performance. How does it sound as a whole? Example: If you're a pianist and worry during your piece about your fingering, you're ignoring the song and how it sounds. Worry about technicalities when you practice. Which should often be.
With time playing in front of and for other people will come much easier. You'll be a natural. So use every opportunity to show your stuff!