This is a modern software application delivery model, in which vendor develop the application, host and maintain the applications by themselves.
Traditional applications are Application service provider or on demand applications. So customer need to pay for the entire application. Moreover customer might not be interested in all the service provided by the software. This create customer a feeling to pay less for the application. All most all traditional desktop applications are on demand or application service provider model.
As web space bringing more attention to customers the idea of software as a service arise. Now web act as a platform to help and enhance service based access to infrastructure, applications and business processes. You can see that almost all modern web 2.0 applications follow the software as a service model. Customers do not pay for the entire application rather than they pay only for use. Even the vendor can expose there service via API which may be web service or REST calls. Since the application resides centrally any patch can be applied anytime without any hurdles. Recent survey says that software as a service and cloud computing holds most strategic values in the coming years.
Let me bring your attention by demonstrating an application. One of the popular platform fastalerts where customers can send alerts which can be sms, email or voice alerts. If I am a customer of the application I don’t need to pay for using application rather than that I pay for the sms. I can purchase sms packages and send from the application. I hope now you got a clear idea what is SaaS?