You've probably seen lots of companies that offer 'domain registration' as a service, but what exactly is a domain, and what's all this about registration?
What is a domain?
Simply put, domains are a way describing someones space on the internet. Much like the old wild west where settlers would pitch their wagons and claim an area of land as theirs, an internet domain marks out a place on the internet as belonging to someone. For this reason they are also know as addresses. In fact, the domain name makes up a part of web sites address as we'll see in a moment.
So what exactly is it?
It's just a piece of text really. A small piece that usually makes sense when read, but it doesn't have to. If you look in the address bar at the top of your internet browser, you'll see something like this

Let's take a look at that in a bit more detail, starting with the .com part at the end. It's a little known fact that domain names should actually be read backwards. In this example the 'first' part of the domain name is .com. This is known as a 'top level domain'. There are over 200 of them, which are meant to describe what kind of content the website at that address contains, or which country it is based in.
Some common examples are
.com - commercial (initially it just included the United States)
.co.uk - a commercial site based in the United Kingdom
.au - Australia
.ie - Ireland
.im - Isle of Man
.org - Organisation
.gov.uk - United Kingdom Government
After the top level domain (TLD) in our example (still reading backwards), we get to the name bit. In this case facebook. Combined with the TLD, this makes up the complete domain name; facebook.com
But what about the www bit?
Another little known fact is that www isn't actually required to make a valid website address. It's more of a convention. The idea is that this last bit of the address (when reading backwards) is like the final step in a list of directions. We know that the domain is a company (from the .com), we know it's called facebook, and the www says it's the address of a world wide web site.
This last part of the address is actually called a sub domain.
Most web sites are actually set up to work without the www and the new trend is not actually use it at all.
Some sites use it to logically split their site up like the news part of the BBC:

Or a developer area and store parts of a site such as:
developer.apple.com
store.apple.com
What is domain registration?
As you can probably imagine, there are quite a few domain names out there. Someone somewhere must have a list.
They do. And because this is to do with computers and is therefore needlessly complicated, there are a number of organisations involved. There is one organisation at the top called ICANN who is in charge. They delegate responsibility of running each top level domain (TLD) to other organisations. For example, Nominet.org looks after all the .uk domains.
ICANN gives companies like 123reg and Godaddy licences to issue domain names on their behalf, who then act as agents, taking your money and the domain name you want, registering it on your behalf with the organisation who looks after whichever TLD your domain comes under.