To make a "real" social networking site you are going to have to throw down some money to buy a domain name and all that boring stuff. Are you looking to do something that serious or are you looking to do somethig just for fun?
What you can do with a school laptop depends how the Internet system at your school is set up. You could probably develop a social site on a laptop if you have the tools to do it but you would have a huge headache trying to run a social networking site that way. Make the site on the laptop then move it to a server.