Intel or Power PC Mac?
intel 2.16 core duo
That's the latest Flash Player that'll work with your OS. It's a zip file. Right click and "save" it. Then quit your browser and mount the DMG & run it.
The bad news is, it won't work with ALL Flash content anymore.
The good news is, having an Intel Mac, you can upgrade to Snow Leopard for $19.99 and it'll run the latest (11.8.800.94) as well as a whole bunch of software the requires 10.6 as a minimum.