A web application is any application that uses a web browser as a client. The application can be as simple as a message board or a guest sign-in book on a website, or as complex as a word processor or a spreadsheet.
A web application relieves the developer of the responsibility of building a client for a specific type of computer or a specific operating system. Since the client runs in a web browser, the user could be using an IBM-compatible or a Mac. They can be running Windows XP or Windows Vista. They can even be using Internet Explorer or Firefox, though some applications require a specific web browser.
Most web applications are based on the client-server architecture where the client enters information while the server stores and retrieves information. Internet mail is an example of this, with companies like Yahoo and MSN offering web-based email clients.
The new push for web applications is crossing the line into those applications that do not normally need a server to store the information. Your word processor, for example, stores documents on your computer, and doesn't need a server.
Web applications can provide the same functionality and gain the benefit of working across multiple platforms. For example, a web application can act as a word processor, storing information and allowing you to 'download' the document onto your personal hard drive.
If you have seen the new Gmail or Yahoo mail clients, you have seen how sophisticated web applications have become in the past few years. Much of that sophistication is because of AJAX, which is a programming model for creating more responsive web applications.