Please check this site, while I am explaining:
Now this is the part of the site that i want to make something alike:

This is a signup form. Now they have spliced the form up in a cool way, I mean if you try to fill something in the fields and press "Fortsæt" (Next in English) then without updating the page, the signup form changes to new fields "Password, birthday, Country" and then the button changes to "Log in".
How have they done that? I suspect they have used jQuery? How should I do this, if I want to do something like this? Please show me with examples as am still learner in jQuery. Thank you.