There are two types of singing auditions: open calls and invited calls. Open call singing auditions can generally be attended by any singer, so long as they arrive at the audition location and sign up before a certain cut-off time. Invited calls are specific invitation appointments for a singer to audition, scheduled ahead of the audition date through an agent or manager. Invited calls often also require the submission of a resume or "demo" (a recording of past solo work by the singer) before being invited to the audition.
Singing auditions will almost always require a singer to prepare a song (or multiple songs). The singer will be encouraged to choose a song similar in style to the singing job for which they are auditioning. For example, if you are auditioning for a rock band, you should sing a rock song. You should sing a selection from a similar musical if you are auditioning for a Broadway musical. Nearly all auditions will require you to bring a piano accompaniment for the song, though some will allow CDs or allow you to sing "a cappella" (without accompaniment). Audition directors also may ask you to prepare a specific song or songs for the audition.
Some advanced singing auditions will not only require you to sing a prepared solo, but to quickly learn or "sight-read" (read new music without any preparation) a piece. Many musical directors will also test your "range" (lowest and highest notes) and your understanding of rhythm. If the audition is for a group singing job, this may be followed by pairing you with other auditioning singers or with pre-selected singers to test your ability to "blend" your voice well (i.e. not stick out).
Singing auditions are typically quite competitive and may draw hundreds or even thousands of singers for very few singing jobs. This can make for long lines and extended waits. Many singers will bring books or MP3 players to pass the time. While some singers prefer to sit by themselves and focus, many others will meet and chat with fellow auditioners. The competition is intense, yet auditions are great places to connect with fellow performers. Either way, being well-prepared is the surest way for you to stay relaxed and ready to go into the audition room.
The people running a singing audition are not the enemy. The majority of them are friendly and professional and aren't waiting for the singer to make mistakes. These people want every singer to walk into the audition room and be terrific. The best problem a producer or director can have is an abundance of great singers from which to choose. Singers make the best impression by being courteous, cheerful and prepared, no matter what. Even if a singer isn't right for that singing job, a good performance and a charming personality will leave a lasting impression that could lead to future work.