Comparing real life with digital, think of a domain as a piece of land. Each domain has to carry a unique name for clear identification. The administration of domains is under the control of Network Information Centers (NIC) of which the top authority is Internet Corporation for Assigned Names and Numbers (ICANN), who assigns further registries, for example on national levels.
Taking this website’s name „creative-dragon-works.com" and comparing it to my other domain „creative-dragon-works.de", you will find the only difference with the ending .com and .de.
These endings are the Top Level Domains (TLD) for which various registries are responsible. There are generic TLD’s like .com, .net, .org, .info, etc. as well as country coded TLD’s like .de for Germany, .fr for France, .it for Italy, .nl for the Netherlands and so on.
In above examples „creative-dragon-works“ is the second-level-domain, which could occur once under each TLD.
Regarding the registry of a domain, in example in Germany the DENIC is the responsible Center. In fact all German Internet Service Providers (ISP) are members of DENIC and therefore in practice, when you have a hosting-account with an ISP, usually you can register a domain directly from there, no matter which TLD you go for.
As you are perhaps aware, there several millions of domains registered world-wide. When choosing your domain name it might happen your choice not being available anymore. So you have to look for another one. Sometimes it helps just switching the TLD.
Example: If „yourdomain.de" is not available, „yourdomain.eu" or „yourdomain.com" still might be free to register. Also changing only one letter or digit could help you out like „your-domain.de".
The only drawback is, that people could mix you up with the other, very similar domain when they surf the Internet.
Regarding your domain, you also wish people to remember it easily! A short name, easy to memorize will perform better in driving visitor traffic to your domain.
As advice when registering a new domain, please be prepared for the ideal name being taken already and have some alternatives on hand.
The choice of the TLD already should be considered carefully with regard to the audience you want to reach. Is your target group a global one, you will try to get .com, .net, .org or .info as TLD. For localized markets it will make sense to head for the respective country coded TLD.