Hi there,
Web services are great for server-to-server communication, but also for client-to-server communication. In the book's example, the browser makes a request for data coming from the server using a web service. You could have done the same using a full postback, but that typically takes more time, and causes the screen to refresh completely. Not so with services. As a very concrete example, consider Google's auto-suggest. When searching on google.com you see a list of potential matches which are retrieved by making calls to the server.
Note, though, that web services are a broad term. For .NET, ASMX services, WCF and Web API would all fall in that category.
There's plenty of stuff written about web services that may give you more ideas:
https://www.google.nl/?gfe_rd=cr&ei=...f+web+services
Cheers,
Imar