Viral marketing is an oft-used buzz word and it may not be entirely evident what it actually means. Viral marketing gets it's name from a virus obviously, so to understand how viral marketing works is to understand how a virus works.
A virus works by infecting a host cell and injecting it's own DNA so that the host cell starts replicating the virus until it can take no more and dies. Then, once where there was just one virus, there are hundreds or thousands, which then go on to infect other cells and replicate again, and again, and again, the virus replicates exponentially.
Like a virus cannibalizing cells, viral marketing works by enlisting your users, either wittingly or not, into promoting your site for you. So that one user enlists 2 more users who each enlist 2 more users who each enlist 2 more users, and so on, and so on. A viral marketing curve would thus look like an exponential curve with every user you get generating more traffic for you.
In a perfect viral marketing scenario each user would generate 2 or more new users. However, that doesn't always happen, especially with the most basic viral marketing applications.
You might in fact already be using viral marketing and not know it. For instance in it's most basic form viral marketing can be represented by an "Email to a Friend" script that lets one user tell another user about your site or product or article.
You can encourage this tell a friend behavior by appealing to the competitive side of a user if your site involves some sort of quiz, test, or game. Once the user gets their score you can ask them if they want to challenge their friends to beat their score and provide them with an email form for letting their friends know how they did.
Another method for getting your users to promote your site for you is with bribes. These bribes do not need to be monetary however. You can run a contest to see who promotes the most new users. The winner or winners could get a monetary reward but also you could simply offer more site features to them. If you run a forum this is very easy as you could offer them things like larger avatars, more private message storage, or other user enhancements. In this way you still get the viral traffic but it does not cost you anything.
If you run certain types of game sites you can provide in-game power or currency to users who do this. Many people game sites actually provide a unique link for each user and anyone who visits that URL ends up providing a power up of some sort to the user who provided the link. This is one of the most effective ways of promoting a new game and can approach the ideal viral situation I mentioned above.
The most successful types of viral marketing though tend to just be things that are either so interesting or so funny or so entertaining that people share them, post about them, and talk about them, so that a massive word of mouth campaign ends up delivering massive amounts of traffic for your site. It is very difficulty to engineer this type of viral marketing though, and it isn't going to be for every site, because it generally relies on some sort of pop culture multimedia content.
Viral marketing is one of the cheapest ways to promote your site, and as such there should be some sort of viral marketing component included with every website, even if it is just an email to a friend form. With other forms of marketing you end up paying for each visitor you get, with viral marketing you only need to get the ball rolling and eventually it could snowball into an avalanche of traffic. Only organic search engine optimization can approach the ROI (return on investment) of viral marketing.
So spend some time and brainstorm ways you can enlist your users to promote your site for you, the beauty of viral marketing is that, if you achieve the ideal viral scenario, you really only one good idea. One good idea to open the traffic flood gates.