Domain names are, strictly speaking, not necessary for the functioning of the Internet. If all the domain names tomorrow disappeared overnight, it would still be possible for computers to talk to computers, by means of their IP address.
An IP address is the four-part number that looks like this: 18.104.22.168
How the System Works
These numbers are what computers use to route data - e-mail, web pages, music files - all over the Internet. But because humans can make better use of names, the domain name system was set up.
When your computer tries to go to a domain that it's never heard of before, it goes through a series of steps. Let's look at the example of hopstudios.com.
First, your computer asks the local Domain Name Server, "Have you heard of 'hopstudios.com'?"
If that computer hasn't heard of it, it will go ask the name server authority, "Who's in charge of 'hopstudios.com'?"
It will probably be told that some other DNS machine at Hop Studios' ISP is in charge of knowing that. So it goes and asks the local ISP's machine, "Hey, what's with this 'hopstudios.com'?"
That machine says, "Oh, I know where that machine is - it's the server at this IP address: 22.214.171.124."
This gets told back to your own computer, and communication between the two can now occur.
Who's in Charge?
It's a complicated question, but most countries are in charge of their own domain name systems. Each top level domain (.uk, .fr, .com, .biz, .tv) has its own rules about who can set up domains within that space and how much it costs.
Ultimately, domain name policy is set by ICANN (http://www.icann.org/), the Internet Corporation for Assigned Names and Numbers. Domain disputes are settled by WIPO (http://arbiter.wipo.int/domains/), the World Intellectual Property Organization Arbitration and Mediation Center.
For more information, check out Wikipedia's article (http://en.wikipedia.org/wiki/Domain_name).
Domain Names Are Not Web Hosting
You can buy a domain name from a registrar and Internet Hosting from a separate web host. If you do, then you have to modify your record with the original registrar which is more hassle than it's worth. So it's a good idea to find a web host that also registers domain names. And these days, that's pretty much everybody. So as you're shopping for domain names, also be thinking about hosting.
NOTE: If you decide to create a blog-style site, you can avoid paying any fees for domain names or hosting by choosing a hosted site. But, the drawback is your domain, or address, will nclude the name of your host. For example: http://mydomain.blogspot.com or http://mydomain.wordpress.com.