Only flash player is required to run a flex projectand also flash project. Flex project also generates *.swf file.
To develop flex project you need flex SDK which is free of cost and open source. You can use Flash builder to build project or you can use any editor to write your code and compile it using flex compiler which comes with SDK.
Server is only required if you want to connect to database.
Hope this helps
There is no special server required. You simply drop the compiled swf file onto whatever server you are running.
The Flex SDK is a free, opensource project:
Adobe of course offers their own pay IDE for it which is Flash Builder 4 (formerly Flex Builder). It is an Eclipse-based standalone or plugin. However, it is of course not required and you can easily get by without it.You can use any text editor, but in terms of a free the strongest option is FlashDevelop which is opensource and runs on .NET. There are free plugins for Eclipse and NetBeans, but they are very weak.
If you are new to Flex and Time is of the essence for your current project. I would highly reccomend getting the FlexBuilder IDE.
No I don't work for Adobe.