There are a number of confusing* terms surrounding your question:
Flash - An interactive content development tool that has lots of vector drawing and animation capabilities, can incorporate interactive elements programmed in ActionScript, and publishes to SWF files.
Flash Player - A browser plugin that runs SWF files.
Flex - A framework for developing rich internet applications by programming in ActionScript and laying out UI in MXML.
Flex SDK - A set of freely downloadable compilers for ActionScript and MXML that generate SWF files.
Flash Builder (formerly Flex Builder) - An IDE sold by Adobe for programming ActionScript code.
Flash Develop - An open source IDE for programming ActionScript.
Long story short, a "Flash game" is a SWF file that runs in the Flash Player.
*Most of the confusion is because Adobe purposely made the terms confusing. Part of the problem is that they occasionally change things, like when they renamed Flex Builder to Flash Builder or when they changed Flex SDK from an IDE to the underlying compilers (and actually I'm not even entirely clear about that last one.)