If you have a web site, you should seriously consider registering your own domain name. A domain name is a name like "thesitewizard.com" or "thefreecountry.com", which you can use to refer to your website. Note that you do not have to be a company or organisation ("organization" if you use a different variant of English) to register a domain name. Any individual can do it too.
The suffix identifies the name as belonging to a specific top-level domain (TLD). As of late 2016, when this article was written, there are numerous TLDs available for general purchase, including .com, .edu, .game, .green, .hiphop, .net, and .org. The most popular of these by far is .com, which is supposed to indicate commercial sites, but in reality has come to include almost everything.
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.
A domain name consists of one or more labels, each of which is formed from the set of ASCII letters, digits, and hyphens (a-z, A-Z, 0-9, -), but not starting or ending with a hyphen. The labels are case-insensitive; for example, 'label' is equivalent to 'Label' or 'LABEL'. In the textual representation of a domain name, the labels are separated by a full stop (period).
HostGator is a company I used for many years with an old firm, and I've found them to be fairly reliable as a rock-solid grounding for a bandwidth-heavy site. Their control panel is also fairly powerful, allowing you to automatically control your fleet of names, as well as locking them down so that nobody can snipe them if you accidentally let a renewal lapse. Their prices are competitive but not the best, but I've found them to be immensely reliable in the past.  Read our HostGator Review.
Not every registrar sells every top-level domain, so if you want an uncommon TLD like .pizza, you might have to try a few registrars. Unusual TLDs often cost more than a standard .com address, but provide built-in memorability and give you more name options, since far fewer .pizza addresses are already taken. You can check name availability at any registrar that offers the TLD you want.
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.

Your domain name is how visitors easily find you online. It is often the first step in getting your website started because it is like establishing your website’s street address on the internet. If you did not have a domain name, you would have to give out your IP address to everyone who wanted to visit your website, which would get confusing and be easily forgotten.
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 john@johnsmith.com 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.