Here is what I did and it worked. I am running Flash CS4, but never installed the standalone flash player that comes with the program.
I solved my own problem with some of the other comments. I used the uninstall_flash_player.exe file from the Adobe site to uninstall any previous Flash Players, then went to the Flash CS4 folders, found the Players folder, opened that folder and found the FlashPlayer.exe file and installed this file. All my problems went away and all of my existing SWF files now play in this player. I don't know if this works for anyone else, but it sure worked for me.
Flash Player is a browser plugin. It's not designed to work as a standalone app for opening FLV or SWF. The standalone player EXE is a different animal.
Folks who don't own Flash Professional aren't going to have a standalone player.
For windows users, the Standalone player installer is in the archived players technote here:
However, it's not an application. It's just an EXE. You have to launch it, then use File> Open to open the local SWF. You can also keep it on your desktop and drag/drop SWF content to it.