The term was originally coined to describe various free email services' practice of appending advertising for themselves to outgoing mail from their users. The assumption is that if such an advertisement reaches a "susceptible" user, that user will become "infected" (i.e., sign up for an account) and can then go on to infect other susceptible users. As long as each infected user sends mail to more than one susceptible user on average (i.e., the basic reproductive rate is greater than one), standard results in epidemiology imply that the number of infected users will grow according to a logistic curve, whose initial segment appears exponential.
More generally, viral marketing is sometimes used to describe any sort of Internet-based stealth marketing campaign, including the use of web logs, seemingly amateur web sites, and other forms of astroturf to create word of mouth for a new product or service.