Flex is a convoluted term for sure. As the history of the product has evolved it has been a j2ee server, an as framework, a compiler and and IDE among other things.
In todays day and age we are at a released version 3, with version 4 in beta. The term flex typically refers to a way of developing flash platform application using the components in the open source framework developed by adobe.
Flex applications are written as a combination of xml markup (mxml) and actionscript much in the same vein that html applications are written with JS. The big difference is that the output of all your mxml and actionscript files is a single swf that can be embedded in a web page and displayed to the world. You can also use Flex to develop AIR applications, applications that can be installed on a users desktop.
The short answer I’ve heard is Flex is “Flash for application developers”. It’s a set of tools that include a compiler and debugger, as well as an application framework and a programming model for building rich Internet applications that use SWF files.
