Getting people to pass along your message voluntarily, mostly through like social networking websites. Another example would be a company making a funny YouTube video and people sending it to their friends raising brand awareness for that company.
Viral marketing is when your site gets promoted exponentially across the web without you having to pay dollar for dollar for each individual visitor you get. For example, word of mouth marketing is the most common form of viral marketing ... people tell people, those people tell people, and the buzz grows exponentially (similar to the way a virus attacks your body).
Setting up an e-mail service using your domain is another example of viral marketing. You would set it up so that all outgoing e-mails have a tagline at the bottom mentioning your product, website, or service. Hotmail, Yahoo!, etc do this. As people sign up for your e-mail service, and send e-mails to their contacts, they all do advertising for you. In turn, their contacts notice the message at the bottom of the e-mails they have received, and then sign up for your e-mail service too. And on it goes, growing exponentially (in theory).