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.
There are numerous domain name registrars. Listed below are just a few, along with my comments, if I know anything about them. Note that the domain name industry is highly competitive, with prices wildly fluctuating throughout the year, every year, so it's impossible to really mention accurate prices below unless I spend all my time updating this page. Please check their sites for the latest rates. All prices below are in US dollars.
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.
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.
The right to use a domain name is delegated by domain name registrars, which are accredited by the Internet Corporation for Assigned Names and Numbers (ICANN), the organization charged with overseeing the name and number systems of the Internet. In addition to ICANN, each top-level domain (TLD) is maintained and serviced technically by an administrative organization operating a registry. A registry is responsible for maintaining the database of names registered within the TLD it administers. The registry receives registration information from each domain name registrar authorized to assign names in the corresponding TLD and publishes the information using a special service, the WHOIS protocol.
Namecheap is a solid choice for registering your domain name. Plus, their site is incredibly intuitive and easy to use, especially on their domain management pages, which can be incredibly helpful. They offer reasonable priced domains, and have a free DNS service, and WHOIS protection. They also offer SSL encryption, for those looking to beef up the security of their domains.
In order to register any domain, you must provide certain personal information that you may not wish to be publically searchable in the Whois database. Signing up for WhoisGuard is a great way to keep your registration data private. WhoisGuard acts as a "shield" for your searchable information, displaying the address, phone number, and email of the domain registrar (Namecheap, e.g.) instead of your own. Keep in mind that WhoisGuard is an optional security add-on that must be renewed separately when you renew your domain.