Hi Luke,
you don't need anything other than to click on 'sign up for free', fill in the application form, slide an arrow, click on 'Sign up' then wait for an email from photobucket requesting that you click on a given link. So easy, but there may be a problem with your age. If there is just ask your Dad if he will open an account for you.
One thing to remember is that as a freebie your upload is limited to 2gb.