3 sites with flash:
Easy downloadFrom: http://www.miniclip.com/Downloads.htm (100+ games on this site)
Less easy downloadFrom: http://www.t45ol.com
If you use Safari as your browser, start the game and then use the "Activity Menu" to find the file ending in ".swf". Doubleclick on it, so the game opens in its own window, and choose "Save" from the File menu.
Some of their Flash files are stuck inside EXE files. This is not at all Mac friendly but there are ways to get around it. My other application, File Juicer, may sometimes extract the SWF inside. It may be encrypted or require special Flash add ons which only work on Windows, so Flash extracted this way may not be completly working.
Nick Jr. Games For kids - ages 0-5 years
Not really made for downloadFrom: http://www.neave.com/games/ (8 classic games)
To save them you need the above tip with the Activity menu, but even then they are not really independent of the network. They have also gotten an "almost full screen" mode in the browser.
More games...Read "Finding Flags Games - 2"
Privacy of Flash gamesWith Flash Player 8 (download page) you can allow downloaded flash games to access the internet. This access is not allowed as default, and usually not needed. It will warn you if the games try to access sites on the web. The games are likely collecting statistics of use for advertising purposes.
You edit the Flash Plugin settings by control-clicking in any of the flash games.