OK, but maybe you got hit with one of Adobe's stupid Flash installer gotchas.
First one: When downloading the .dmg installer, you may get a message that it was installed!!
It's NOT!!! The .dmg installer is just downloaded and "installed" to your default Downloads folder, usually the Users Download folder or Desktop.
Second: When you locate and open the flash.dmg installer, you must click the very small and almost camouflaged approval checkbok and then click the Install button.
It will tell you to quit the browser you use if it's open, and just follow the directions and carry on.
That should get it installed and working for you.
PS & BTW: 2GB RAM is the minimum I'd want to try using, even with SL 10.6.8, and I'd suggest installing more.