GoDaddy — This extremely popular registrar (probably the biggest today) offers .com domain names for $9.99 (plus 20 cents) per year (or $6.99 plus 20 cents if you transfer from another registrar). They have a web interface to manage your domains, free web redirection (where people who visit your domain will get transferred to another web address of your choice), free starter web page, free parked page or free "for sale" page, and an optional private domain registration where your domain is registered in the name of a proxy company. Besides .com, they also sell .us, .biz, .info, .net, .org, .ws, .name, .tv, .co.uk, .me.uk and .org.uk, etc. Note that (as with all registrars) the exact price varies depending on which domain you are registering (for example some domain suffixes are more expensive than others). Both credit card and PayPal payments are accepted.
For the most part, you can pick whichever TLD you prefer, though a few have restrictions -- you can't buy a .edu domain unless you run an accredited post-secondary school. Many, but not all, country-specific domains (.us, .uk, .jp) are restricted to citizens of the country in question. Unrestricted country domains are often used for "domain hacks," where the extension becomes part of the title, such as YouTube's alternate address, "youtu.be."
Who is in charge of all of these processes? Most countries manage a Network Information Center (NIC). An NIC is the main database that has all the information regarding what domain names have been registered and the IP address associated with each of those domains. The nonprofit organization the Internet Corporation for Assigned Names and Numbers (ICANN) is in charge of creating and managing the procedures for internet namespaces.
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.
If you’re still with me then you might be wondering who came up with all of this. The answer is ICANN — the Internet Corporation for Assigned Names and Numbers. ICANN is a non-profit whose role is to coordinate all the names and numbers that keep the Internet online. They outsource a lot of the heavy domain name lifting to registries like Verisign, who pay ICANN for the privilege.
Premium domains (also known as aftermarket or pre-registered domains) are short domains, often just one word or even just 3-5 letters. Most of them have a .com extension but many premium domains end with .org, .net, and .biz. These domains include common words and are generally the most memorable. Because companies value short domains that match their company name or products, these domains are typically the most desirable. Additionally, certain domains sold by different registries are considered premium and therefore have a higher price point. In some cases, the renewal costs of these higher-priced domains are also quite expensive. Some high-priced premium domains, though, renew at a regular (lower cost) rate, giving you a better value in the long term. Make sure to research the overall cost of the domain plus renewal to find the right domain for your budget. You can browse Namecheap's premium domains in our Marketplace.