Intel or PPC Mac?
>About This Mac will let you know.
Download the latest working version for OS 10.5.8 (Intel) here: Flash Player 10.3.183.90
Unzip it. Inside thereshould be a few folders. One is named "10_3_r183_90".
Inside that folder is a flashplayer_10_3r183_90_mac_intel.dmg
Double click that and it will mount on your desktop, and open a window with the installer.
Right click the installer and choose "Show Package Contents"
Go to: Contents/Resources and double click the Adobe Flash Player.pkg file
The installer will open, ask for your Admin password, and it will install.
Then, since your Mac is Intel powered, I recommend upgrading it to OS 10.6 (Snow Leopard). Apple sells it for just $19.99 and they ship free. You'll be able to run a whole world of software that 10.5 can't handle, including the latest Flash Player.