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.
If you ever change your web host (the actual physical location of your website files), your domain name goes with you. Your regular visitors or customers who knew your site name as (say) www.thesitewizard.com would not have to be informed about a change of web address (also known as "URL"), since as far as they are concerned, the site is still at the same place. They would simply type your domain name and they'd be transparently brought to your new location.
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.
What’s easy to forget, though, is that Verisign have managed .com for the past 20 years without so much as a hiccup of disruption or downtime, and that there’s no guarantee that a more democratic or competitive process would have resulted in the 2 decades of .com stability that we’ve enjoyed. Internet governance is obviously important, but we should remember that the overwhelming majority of users don’t know or care what’s going on behind the scenes.
Absolutely. Owning 'yourname.com' (as well as related domains such as yourname.tech or yourname.me) is a great way to brand yourself and retain control over your name's online presence. With a personal domain name, you can set up a portfolio, blog, or hobby site that’s associated with your own personal life. You can also set up a custom email address like firstname.lastname@example.org allowing your visitors an easy and memorable way to reach you. So even if you have no immediate use for yourname.com, it's wise to register a personal domain to ensure that you (and not some stranger) control your name online.