This shift in thinking is not only limited to the web; desktop applications can also leverage these ideas. Locally running rich client desktop applications can be enabled to expose their features to other local, or even distributed, applications.
In fact, in Unix, this concept has been around for decades in the very simple, yet immensely powerful, form of software pipelines. By chaining the output of one program's data (via stdout) another program, with it's own distinct functionality, can then manipulate the result. This simple technique has enabled Unix developers and administrators to solve an infinite number of problems from a small set of tools.
Between these two extremes is the idea of desktop mashups. But instead of following the pull nature of the web, or the push nature of software pipelines, desktop applications can take advantage of the event driven nature of GUI systems and thus create a rich desktop experience. Thus separate programs, with distinctly different functional offerings, can leverage each other to provide functionality beyond the others original intention.
For clarification consider this simple example: you have two separate applications: one which rips DVDs, another which catalogues them ready for playbook (e.g. iTunes). When the DVD ripping applications completes it sends a message to say a new movie is available. The cataloging application - which has no direct knowledge of the first, receives the event and interrogates the new movie and adds it, and its metadata to the catalogue.
Of course this is nothing more than Event Driven Architecture which is hardly a new concept. The move to mashups comes when we create new applications from the existing apps functionality. So, in our simple example, we create a new application which fetches film reviews from imdb. When the user inserts a DVD from ripping it fetches the review so the user can decide whether they should go ahead. When the user does rip the DVD the review application updates the catalogue application so the review is part of the movie's meta.
Enterprises are ripe for desktop mashups. A significant number of problems for users is the lack of integration between applications, whether they are different internal apps or even third party. Another advantage, especially in large enterprises, is that development teams can focus on developing small applications that do their jobs well, or integrating with legacy apps, and mashing them together rather than struggling to develop all-in-one god apps.