A simple explanation: A domain name is an internet web address that begins with www and ends with a suffix e.g.: www.example.com
Essentially, it is an "identifyer" and allows people to find you on the internet.
E.G. www.purpledogdesign.com - that's our domain name.
Domain names can have many different suffixes, (called G/TLDs) e.g. .co.nz, .org, .net and so on
For a complete explanation: http://en.wikipedia.org/wiki/Domain_name