You will do the initial online application, submitting your resume, essays, and additional information, plus have to get 3 letters of recommendation submitted.
Later, you will be contacted by a regional recruiter who will do an in person interview. A lot of the questions are about your background, why you are interested in joining the Peace Corps, your thoughts on living abroad, volunteer service, etc. Pretty basic. I was asked to describe a lot about my previous abroad experience, and also a lot about my background in health (I said I wanted to be a health volunteer). They ask you several times about your commitment to serve abroad, away from family and friends, for such an extended amount of time.
If you do well on the interview, you will be nominated for a position, and then you have to pass medical and security/background clearance. Make sure you do all your appointments as quickly as possible and submit everything as soon as possible so your application can keep rolling. A lot of people get stumped here. It's easy to get done, you just need to turn everything in without errors, and follow up with the DC office often.