What is a domain name
There are two types of "Domain" Names, which can get confused. Usually the term refers to an Internet Domain Name, but sometimes can refer to a Windows NT Domain Name.
Internet Domain Name
Internet Domain Names were developed as a way to easily identify servers on the Internet. When the Internet first started these domain names, and their related IP Addresses, were all saved in a single file (called a hosts file. There is still one on every computer) that resided on every single computer on the Internet. This was fine when there were only a handful of computers, but by the time the number of servers reached the thousands, keeping these files in sync rapidly became a nightmare.
In order to make the management of these host names easier, a distributed name serving system was developed. This became known as DNS. Looking up information in this database is one of the reasons we have the dots in domain names.
Every dot in a domain name is a level separator. Reading from right to left, we have what is refered to as the Top Level Domain, or TLD. This is, for example, the .com or .uk of your domain name. When a computer is trying to contact your web server, the first lookup is the TLD. This then refers your computer to the next naming server down the tree, until it reaches the last level which should have the IP address of the server in question.
TLD's were given to every country in the world, and because national TLD's are all two letters (.uk, .us), some interesting coincidences appeared. For example, .tv has nothing to to with television. It is, in fact, the national TLD for the Tuvalu islands. The 4000 or so islanders have turned this coincidence into a cash crop.
There are also a number of non-geographical top level domain names. The best known of these are .com and .org. You can see a full list of TLD's here.
When you acquire a domain name you do not own it. It is more like a lease. It can be taken away from you if you are infringing someone's intellectual property, or abuse the Internet by sending spam, or hosting web sites with illegal content. Every TLD registry has it's own terms and conditions. Nominet, who manage the .uk namespace, has theirs here.
Below your domain name you can have any number of sub domains. Even sub domains can have sub domains. Technically there is no limit to the number of host names and sub domains you can have of your domain name, until you reach the maximum host name length