Even when you register and pay for your domain name, you won't necessarily be able to use the name for several hours or even a few days. The domain must propagate, meaning that the official domain name registry must be updated with your website's Domain Name System information. That's something that occurs on the backend without any need of input from you.
Keep in mind, however, that free domain names are usually free only for one or two years, after which the registrar will bill you for the annual or biennial fee. In other words, the provider of the free domain name pays only for the first billing from the registrar. Also take note of whether or not the provider charges a fee for setting up a domain name. Most services offer to transfer an existing domain name to their servers at no cost, but sometimes you'll find a setup fee over and above the registrar's fee.
My personal preference is to register the name directly with a domain name registrar rather than through my web host. I've heard stories, in the past, of less-than-reputable web hosts that registered the domain under their own name, making them the owner of the domain rather than you (although I don't know if such web hosts still exist today). Registering direct with a domain name registrar allows me to make sure that I am registered as the owner, the administrative and technical contacts. Being the owner is vital — if someone else places himself as the owner (such as your web host), he can always decide to charge you some exorbitant fee for the use of the name later, and there is little you can do. The various other contacts are less vital, but may still play important roles, depending on your registrar. For example, for some registrars, the administrative contact's approval is required before a domain name is transferred out of a web host (or at least, it used to be). If he/she cannot be contacted, the technical contact is used.
Next, someone has to tell the rest of the Internet that google.com points to 220.127.116.11. This is done by a “registry”, which is also the wholeseller that provides domains to registrars. Each top-level domain (TLD) — .com, .net., .org etc — has a registry that manages it. The .com TLD has been managed by the same registry since the beginning of time — an extremely profitable monopoly called Verisign. More on that later.
There are tons of great domain names available for registration and it can take as little as five minutes to register one at any domain name retailer. These online stores make it very easy for you to search for available domain names with name suggestion tools. They also offer complementary services, such as Web hosting and website design so you can register your desired domain name and set up a website all at once.
Administrative contact. A registrant usually designates an administrative contact to manage the domain name. The administrative contact usually has the highest level of control over a domain. Management functions delegated to the administrative contacts may include management of all business information, such as name of record, postal address, and contact information of the official registrant of the domain and the obligation to conform to the requirements of the domain registry in order to retain the right to use a domain name. Furthermore, the administrative contact installs additional contact information for technical and billing functions.