Familiarize yourself with the product. Watch Nickelodeon's programming and make sure that this is the sort of acting you want to go into. Nickelodeon is geared towards children and, as such, their shows tend to be more lighthearted and less dramatic.
Get as much acting experience as you possibly can. This doesn't necessarily have to be anything grandiose. School plays and community theater are all good places to start. It will also give you knowledge of the auditioning process. Take acting classes to improve your skills.
Look for a reputable talent agency and apply. You will need to send them a professional head-shot photo and resume, along with a cover letter. If accepted, let them know that you are interested in auditioning for Nickelodeon. Talent agencies are constantly informed of any opportunities and will send potential actors and actresses who fit the bill.
Know what to expect if you are sent on a Nickelodeon audition. Introduce yourself and be professional. While the casting director may ask you to recite a monologue, in most cases you will be asked to read from a script. You won't be given much time to look over it, so give it a good glance and begin. You may also be asked to read with another person. Afterwards, thank everybody and leave. It will win you no points to linger around or ask a lot of questions. Remember that they have many other people to see.
All there is to do now is wait. If you are called back, they may very well be considering you for a role! The casting director will be giving you more direction during this call-back audition, so pay attention and do your best. Who knows? You may soon be Nickelodeon's newest star!