SaaS is the concept of the Global Village
SaaS is any software application that you run that is not located on your premises. It is a full-blown application, not a component part of something else. It is not a way to build applications. It is not a plug-in to other applications. It is never something that sits on your machines. Instead of having the application running on servers and data storage in your data center, it is running in the vendor's data center.
Software is not the program but the service
The way SaaS applications are licensed is different from on-premise applications. Instead of buying the license to use the application, and then paying for software maintenance to support it and keep it current, you "rent" the software over a period of time - usually monthly or yearly. Instead of buying and installing infrastructure and then paying ongoing operating and maintenance costs, the vendor runs the application on their infrastructure.
Cost is not the matter of worry but the opportunity
The cost of the SaaS application covers the costs of the software itself and the ongoing operations and infrastructure costs. When you run a SaaS application, you generally log into your vendor's web site and you are on. You can say that SaaS applications are running "in the cloud," and you would be correct.