Visit the "Casting Calls" section of MTV's website.
Scroll down the list until you find a show which you would like to apply for and click on it.
Read the casting call and check if you fit the description of the people MTV is trying to cast for this particular show. On the casting call page, producers of the show usually list certain requirements, such as the age, personality or gender of the people they are trying to cast.
Apply to be a participant on that show by following the application process outlined on the page. Most MTV casting calls require you to send an email to a specific address and to attach one or several pictures of yourself. However, some shows may require you to send additional information, such as your motivation for being on the show.