There may be variations among private and independent schools, but no matter the paperwork or the process, schools want to get a full sense of who your child is before granting admission. There will certainly be a comprehensive application form to complete, but additional material could include at least two or three years of your child's previous report cards, a confidential report from your child's current school, medical information if applicable or letters of recommendation. Depending on your child's age and the school, it may be necessary for your child to write the SSAT or some other entrance exam. And many schools have an interview process with you and your child. Schools you're interested in should be able to provide a clear outline of what their application requirements and process.
Learn more about the application process here