Moving website from the old domain to the new one

Moving website from the old domain to the new one

I'm a new in web development so the question is maybe pretty simple, but I
can't handle it by myself.
I have a wordpress website which hosts on hostmonster.com and it has a
URL, let's say, http://aaa.com/bbb.org. So it is located in the
subdirectory bbb.org of the main directory http://aaa.com in my host
monster account.
I also have a domain name, let's say, http://bbb.org on godaddy.com and
now I want to make my site available on address http://bbb.org. How to do
that?
I did the following thing. I forwarded http://bbb.org to
http://aaa.com/bbb.org in settings on godaddy.com, but it's not what I
need, because user still see old address (http://aaa.com/bbb.org) in his
address line. So what am I doing wrong?