Imagine having in your company 2 or 3 applications that "kind of" do the same thing, maybe say, both get information from your clients, one gives you his basic contact information, and the other one more detailed contact information.
Then you decide to add to your client a new element of information, DOB for instance, and are required to return the "age" of it based on the DOB.
Your developers will now have to not only modify two applications to accommodate for the new field, but also add logic into each application, to calculate this value.
Enter Web Services
You grab your developers and ask them to create a "centralized" way of getting that information so you don't have to modify the logic of your programs every time there is a change of this nature. They in turn create this "service", which, when queries, would return anything you want both your applications to process.
Your calling applications become leaner, because you offset all the fuzzy business logic from your programs and stick it into one service that will always provide the latest, up-to-date information of any client you choose.
Internally, your applications will query this service and establish a connection with your central web service, without going into details, this central web service will return the needed information (with the age calculated) using a predefined format that would be consumed by all the applications in your firm.
Web services can be extremely complex, but once in place, can be consumed by a wide range of devices, your phone, a web page, even real speech (Siri?)