Macromedia Flex is a standards-based programming methodology and runtime services for developing and deploying the presentation tier of applications that combine the richness of the desktop with the reach of the web: Rich Internet Applications (RIA).
http://www.adobe.com/resources/business/rich_internet_apps/

They are developer tools, libraries and runtime services that will enable developers everywhere to build and deploy rich Internet applications that take advantage of the Flash Player runtime

It combines a powerful Eclipse-based IDE, and a rich set of J2EE-based runtime services, and provides developers the most comprehensive solution for delivering cross-platform Rich Internet Applications available today. And recent releases of technologies like the Flex-Ajax Bridge and Ajax Client for Flex Data Services under open source licenses allows incremental addition of Flex components to existing websites and applications, whether they are based on static HTML or Ajax technologies.

Flex helps organizations:

• Boost ROI by improving end-user experiences
• Leverage existing tools, workflows, and infrastructure
• Deliver enterprise-class Rich Internet Applications

Rich Internet applications offer benefits for:

Business managers

• Increasing the number of successful end-user transactions
• Helping your online business get to market faster
• Keeping your application development and deployment expenses down

End users

• Reducing frustration through an interactive and real-time user experience
• Shortening the learning curve for new and novice users
• Saving time with a single screen interface