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.
Namecheap — This company provides .com domains for $10.69 (plus 18 cents) per year ($9.69 if you transfer from other registrars). Along with your domain, you get free email forwarding, free web redirection (where anyone going to your domain is automatically directed to another address of your choice), free domain name parking, etc. For the first year (or at least, at the time I checked their prices), you can also have their WhoisGuard (where your particulars are masked from public view) for free. They have a wide variety of domain name extensions available, including .net, .org, .biz, .info, .us, .co.uk, .co, .de, etc. You can use either a credit card or PayPal for your purchases.
It's good to decide early on if you want to have a brand domain or a keyword domain. The benefit of a brand domain is that your domain will most likely be considered as trustworthy, and immediately shows visitors that it's linked to your business or personal blog. A keyword domain, on the other hand, has the advantage that it will be better optimized for search engines, and therefore will get higher search results. In recent years, however, the drastic improvement in search results thanks to a keyword domain has begun to decrease.
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.
FatCow has one of the cheapest offers for an entire website package, with hosting and domain names for just thirty cents during your first month. After, that the price is $10.99 a month or $59.88 for the whole year, in a shared hosting package with enough bandwidth for most people. This is a limited promotion I found, but they still sell fairly well-priced packages (their "fat" options come with extras like marketing and advertising credits), and will gladly offer you perks for transferring your domains and hosting to them.
Note that you can also transfer your domain name from one registration service to another. You'll want to do this if you're not satisfied with your current domain hosting service, if you find a better deal when your current registration is coming due, or, most likely, if you've signed up with a web hosting service that will also transfer your name to its site. Expect to get the transfer for free, but if that isn't offered, search for another domain hosting service.
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.