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.
Although Adobe produces an IDE to help in developing Flex applications it is not a requirement. Everything