For the regular Flash 9 browser plugin, please see RestrictedFormats, RestrictedFormats/Flash and its "free alternatives" section.
Since Adobe Flash player 9 beta, Adobe has released a standalone Flash Player (Projector) for linux. Prior to this the only way to obtain and standalone player was to use Wine - this is no longer necessary.
Installing the Linux Version
The standalone player (Flash Player Projector) can be downloaded from the Adobe website (the page links to Windows, GNU/Linux and Mac files for the plugin and the standalone player - look for "Download the Linux debugger and standalone players").
Unpack the archive you downloaded and navigate to the folder release (via the standalone folder). Inside this folder will be another archive (flashplayer.tar.gz), so extract this and there is your standalone player (double click to launch).