A domain name makes your website easier to find, and it does not change when you change website hosts. You can also use it to protect your name, buying the domain of your name so it is not bought by an enterpriser for purposes you might disapprove of, such as a porn site.
If you have a business, it makes you look professional. Also, if the domain name is catchy, it can attract repeat customers, while catching the eye of first-time customers. Your website will be easier to find and attract more people, particularly if you choose your domain name well. Depending on your host, you can also get an e-mail with the domain name, which also assists your business and its apparent legitimacy.
A legitimate-sounding domain name will increase people's trust in your website; meaning, if you are a business, more people will be willing to buy from you. It tells customers that you are serious about your business. The domain name even acts as a form of advertisement, advertising your brand, purpose, or product, if you choose wisely.