Web service application

I have probably stupied question and simple, but i can't really find anything about that on internet.

Let's say I have an application for bookkeeping records, and I want to I host the application to enable the user to register and to pay usage on a monthly basis, but that every new registered user, you probably need to get a new instance of whether the database in order to data were each protected by another user?

Now, I do not understand how these apps, they call it? Web service applications or how? and say that terminology is necessary to search the Internet to know where to go and take a look at how to do something like that?

I sincerely apologize for this kind of question, but I really need some help because I do not know where to start.

Thank you for your understanding