I own a website and it is completely made out of html and other file extensions.
But I got asked to make a login/signup, I don't know how to make the login form store the data and send a confirmation email to the people before they can access their account?
In order to do this you will need to have some programming running on the server. If you don't want to learn to do that yourself, then there are a number of free systems available that will help you. Wordpress is a good example. You can add plugins to it to get it to work in the way that you need. You can find out more, and download it, at: http://www.wordpress.org/
There is a directory of plugins at: http://wordpress.org/plugins/