Viral growth refers to dynamics built into your app usage that naturally foster an ever greater number of signups. This could be an increasing number of signups via referrals, greater traffic because of inclusion in Twitter lists that are in turn broadcasted in multiple channels, etc.
Network effects on the other hand relate to competitive advantages gained by adding people or companies to your platform or service. A network effect of Android, for example, is that a greater number of devices entices a greater number of developers to program in the ecosystem.
And now the eternal question: which one should I seek? Well, ideally both, however don’t feel pressed to change the nature of your product to accomodate an ever-growing number of buzzwords. A great source for network effects and viral growth ideas is the Business Model Generation book. Happy reading!