Domain names are a convenient way to remember website URLs. Computers find websites using IP addresses - a series of numbers separated by dots. People find it much easier to remember names and words rather than numbers, and the domain name system exists to convert between the two. For example, it is quite easy to remember google.com.au, but not at all easy to remember 64.233.167.104 - but if you enter that into your web browser you will still end up at Google.
All serious websites will have their own domain name. With over 10 billion pages now on the web, search engines are increasingly ignoring any "websites" that are hosted in the free webspace provided by internet service providers, and listing only sites which have their own domain name. Furthermore, people experienced in using the web will often also ignore sites that do not have their own domain name, as most are simply not serious websites.
A domain name short be short enough to easily remember, but long enough to be descriptive. In general you should also avoid hyphenated domain names except in special circumstances.
