How can I run Flash applications on iOS?
On a non jailbroken iDevice, your best options are either the SkyFire browser or the iSwifter app. SkyFire works only for video and iSwifter does work for some Flash content, but neither is a perfect solution, in my opinion. (iSwifter tends to have some funny browser behavior when it comes to touch input.)

Alternatively, you can install "Frash" on to a jailbroken iDevice. I've had varied success with this, however. Some flash apps work fine, most work to some degree, and then some don't work at all. There's no real guarantee what will or won't work with it (just as there isn't a guarantee of what will or won't work with iSwifter or SkyFire).