Web Hosting and Domain Names are technically two different worlds.
A web hosting company stores or hosts your website. They have servers that store your files like Joomla, images, etc.
Your domain name is part of a national registry. All web hosting companies can sell you a domain name. With some companies, like Host Monster, you get a domain name for free when you sign up.
You can purchase domain names at many places on the web. In the end you point the domain name at your website files on the server at your hosting company.
For example, you could purchase dogs.com at Go Daddy, Cats.com at Modwest.com and then point those names at your website hosted at Host Monster.
Domain names are paid for annually while web hosting is a monthly charge.
It is important that you keep your domain name login information handy and in a secure place. One of the number one problems I see is people buy a domain name, pay for it for a couple of years and then when it comes time to renew they can't find their login information. Another problem is that the credit card information on file has expired so the automatic renewal doesn't work.
You want to be able to renew your domain name immediately because if your domain name expires. Someone will buy it immediately and then sell it back to you for a lot of money.
Another reason to keep your login information secure and in your possession is the hassle it is to regain this information. It's like bank account security and taken very seriously. If you loose the information and you cannot reset the password because of a bad email address. you have to prove that your the owner of the domain name. This can be quite a process involving the following: go through the ICANN UDRP process and provide a ruling affirming your ownership, you have to go through legal channels and provide a court order affirming your ownership, and wait for the domain to fully expire and register it as new once that happens.
I have seen a loss of login information when there is turnover at businesses and nonprofits. Someone buys the domain name and then moves on. Two years later everyone is scrambling trying to figure out who bought the domain name, and what email address and credit card they used.
You can find basic information about your domain name by using www.WhoIs.net. Whois.net will provide you with ownership information unless you paid extra for your domain name registration information to remain private.
Here is what the whois information looks like for Irestone Web Design.