RIA stands for "Rich Internet Application". A rich internet application normally runs on a web page but provides the same functionality as a normal desktop application.
Web applications have traditionally been hampered by what they can do on the user's machine. Applications that try and interact with users using static html pages, need to constantly talk to a web server to manipulate data and create new web pages for the user. Performance of these applications is always dependent on the number of round trips needed to the server and how fast the new pages can be created and displayed to the user.
A rich internet application tries to lower the amount of round trips to a server. They use a runtime engine that is downloaded to the user's machine. The engine enables the application to handle all the interaction with the user. It only talks to an application server when necessary.
RIAs also provide a richer set of features for the user interface such as animation or the ability to drag and drop.