Adobe abandoned Flash for Linux years ago. Which is a good thing because Flash is a security nightmare.
If you really must play a flash video, the best method is to just download it and play it with mplayer or similar. Or convert it from
.mp4 or something with
handbrake. There are several video-downloader plugins available for firefox that make it easy to download a video rather than play/stream it.
Alternatively, use google
chrome or the open-source version
chromium with the Pepper Flash Player. Note, however, that flash is and always will be a constant stream of security holes that may be used to compromise your system via a cunningly mal-crafted flash program.
I have no idea if it's pre-packaged for Centos or not, but it's packaged for debian as
pepperflashplugin-nonfree - that contains a script which downloads google
chrome, extracts the flash player and installs it for
chromium to use (so you can get the flash player in chromium without google's other proprietary spyware).
Note that support for 32-bit has been dropped by chrome's pepper flash.