Flex is an application framework that allows developers to easily build mobile applications for iOS, Android, and BlackBerry Tablet OS, as well as traditional applications for the browser and the desktop using the same programming model, the same tool, and the same code base.
Flash Builder is an enterprise-class Eclipse-based IDE to develop Flex applications. Flash Builder includes state-of-the-art productivity features (code assist, refactoring, etc), on-device debugging, and mobile simulators that allow developers to test their applications on different screen sizes and densities. Flash Builder allows developers to package native application files (ipa, apk, bar) for release on the Apple App Store, Android Marketplace, and BlackBerry App world.
The Flex SDK contains hundreds of out-of-the-box components. DataGrids, Charts, Formatters, Validators, and numerous other UI controls are the building blocks for applications of all sizes. Components can be styled and skinned to fit the look and feel you want. There are also hundreds of third party open source and commercial components available for Flex. Mapping APIs, Data Visualization libraries, and Cloud APIs provide the building blocks for assembling great applications. Check out Tour de Flex to see many of the available components for Flex.