The main advantage is the separation of functions. E-mail, like free web-space, is used by some ISPs to help tie in customers to that ISP. When the ISP becomes ropey users are reluctant to move because they have to change e-mail address, web-site host etc. as well as ISP.
There are many sites you can use to check and register a domain. Try
www.godaddy.com.
edited to add:
RobertoS mentioned Purple Cloud who may be perfect for your needs.
Edited by deleted (Thu 31-May-07 20:39:36)