On my website, how can I have something where people can register with a username and password, and then log-in with it to view files that un-registered users can't?

I don't mean just password-protecting files, but like someone who's not logged in/doesn't have an account tries to click on a link, and they get prompted for a username and pass, and they enter it, and then they don't have to enter it again until they log out, or their session is timed out.

This can be done using cookies or sessions. Which language do you want to do it with ? PHP or ASP or what ?

http://hotscripts.com/

As nakulgoyal said, this can be done easily with any server side scripting language such as PHP, Perl or ASP. Hotscripts has a bunch of them.