A domain name is the address that users will enter in their web browser to access your website. To get to this website from an empty browser screen, you would have to type find-me-free-web-hosting.com. The domain name postion of the web address you enter is case insenstive, so find-me-free-web-hosting.com will take you to the same place that FiNd-Me-FrEe-WeB-HoStInG.com.
The .com in our domain name is the top level domain (TLD). TLDs are administered by specific organizations. For example, the .com TLD is administered by ICANN.
It is likely that you will never deal with the organizations that administer your TLD. Registrars have the authority to register domains and intergrate with the controlling organizations. You will deal with domain name registrars for all but the most problematic situations. There are hundreds of registrars on the net that will surely try to get your domain name registration business.
If you do not already have a domain name when looking for hosting, be aware that most web hosts will offer to register your domain name for you as part of the account opening process. Most will even offer initiate the domain name transfer process as part of your move to their hosting if you choose (this is almost never required as domain name registration is a completely separate process than web hosting)