A domain name is essentially a signpost on the Internet. Almost every website you've ever been to, and every email you've ever composed, has used a domain name in its address.
People register domain names in order to 'stake a claim' to a particular name — whether for business or personal reasons. Once you register a domain name, it is entirely yours, no other party may use that identity online as long as you continue to pay the yearly renewal fee and abide by the terms of use.