Note that this does not give you your own domain name. They just add a record in their own DNS server that adds a machine name to their domain name and maps it to your IP address.
ftg explained this very well. I will add that you can register a domain at any registrar, and your ISP will probably sell you a DNS hosting service so your domain name will map to your IP address. However, these services are usually aimed at businesses and bundled with web hosting, so it probably won’t be as cheap as you want.
Remember that you will also need to run a web server to host your own web site, and an SMTP server if you want email.
Although all posters have given good information, I have a funny feeling that you want a domain name but you don’t know exactly what you want to do with it and therefore may find this information confusing. Let’s start from the beginning–why do you want to do with your domain name?