Adobe and Apple are "in discussions" about bringing the Flash Player to iPhone but so far nothing. I've got to believe that it's politics or money preventing it happening, and not technology, as Adobe have demonstrated Flash player running on the iPhone.
Those phones that do have Flash player (i.e. not iPhones) have something called Flash Lite. There have been three major versions of Flash Lite, unsurprisingly called 1,2 and 3. As the processors in mobile handsets have become more powerful Adobe has added functionality in each version. Flash Lite 3, though, is the only version that supports video. Many new handsets coming out have Flash Lite 3 installed.
You can read more about Flash Lite and the handsets it runs on at
http://www.adobe.com/devnet/devices/
