RichInternetApplication (RIA) is a WebApplication that have extended UserInterface capabilities, approaching the functionality (UI wise) of a ClientServer application.
RIAs typically transfer the processing necessary for the user interface to the Web client but keep the bulk of the data (i.e., maintaining the state of the program, the data etc) back on the application server.
Typically RichInternetApplications are implemented in JavaLanguage, DotNet or MacromediaFlex using ExtensibleMarkupLanguage to transport data. However, other possibilities like GoogleMail? can be similarly categorized (see http://www.infoworld.com/article/04/10/22/43OPstrategic_1.html).
(unsure of accuracy of info in rest of paragraph) The extra functionality is achieved through UI engine client-side code being passed to the user machine in the background, thus extending the capabilities of the browser.
ActivexTechnology from MicrosoftCorporation has been delivering similar extensions for a number of years. However, that technology is specific to the users of Microsoft OS, and require registration of the ComComponent on the client, and was then beset with deployment problems (e.g. maintenance and upgrade issues).
RichInternetApplications have had a long history of existence but stayed at the fringes due to complexities in management (e.g. state). See "The state of rich Web apps" at http://www.infoworld.com/article/04/11/05/45OPstrategic_1.html.
Revival of interests in RIA also paralleled the increased interests in using XML in novel ways to enhance WebApplications.
AJAX - AsynchronousJavascriptAndXml? - AjaxWebApplications
See http://en.wikipedia.org/wiki/AJAX at WikiPedia for definition.
Some "rich internet applications" are not really internet applications, just fat-client browsers or addins that use the internet. I think stronger criteria is needed to avoid abuse of the term.
See also: WebFormMethodologies, WebBrowserMissingWidgetWorkArounds, ThinClientHasFailed, RemoteGuiProtocols