There are financial data processing applications written in object-oriented Java which have nothing to do with the internet or Web. Executing on an EJB-based platform or as standalone applications executed from a command-line.
I've worked with Java for the past ten years. I did several projects, most of them for large companies here in the Netherlands.
My previous project was for the Dutch railways. They are busy building a new traveler information system. One part of that is that they will replace all the old-fashioned indicators on all the stations with electronic screens, which are ofcourse much more flexible. To feed the new screens with information about what for example the next train is, when it will arrive, etc. a new system was built, in Java, that receives information from a number of back-end systems, such as the traffic control system, train planning system etc. and that transforms all that information, according to a large and complex set of business rules, into messages that are sent to the screens. It was an interesting project and one thing that's fun about it is that the result will be visible on all the train stations when it's in production.
I've also worked at an energy company. A few years ago, the energy market here was liberalised, which means that everybody can now choose his or her own energy company. To make it possible for people to switch from one energy company to the other, you'd call your new company and tell them you want to switch to them. It's then up to them to contact your old company, find out what your contract is and other information, and then switch you to your new company. The largest part of the switching process is done automatically. A protocol was invented that defines a sequence of messages that go back and forth between your old and your new energy company, and we had to write software that implements this protocol and makes the necessary changes in a number of administrative systems of the energy company.
At the moment I'm working at a company that sells a content management system (software that companies use to maintain their websites). They want to integrate a third party product with their software.
I've also worked on a number of smaller projects, mostly web applications, most of which are for internal use inside companies. I've also worked on a big Java desktop (Swing) application, also for internal use inside a company.
Java's strength is mainly in enterprise, server-side software. Java is not used very much in consumer desktop or mobile applications.