This is a public service. I don't usually write about these things, but I figured people could use some useful information. The information I found wasn't up to date. This is what I know as of 29 November 2012. If a couple months (or years) have passed, be sure to verify for yourself.
Longer story: Our kids' school got a great donation of iPads for the teachers and enough extras for one class to each have one (for the day / hour). However, one of the teachers expressed frustration that the Flash-based video about the Mayflower on Scholastic's website wouldn't work - because Apple's products will NOT play Flash.
I assumed that it was a lost cause, but I figured it couldn't hurt to check. It turns out there are several alternatives - essentially alternate browsers to Safari that have the ability to run varying levels of Flash. I was looking for Flash video, not games or other heavily interactive capabilities. Wired Educator provided a nice overview and several recommendations, including some that are no longer accurate as indicated in the comments of the piece - and as I discovered in testing the free options.
I tried Opera Mini Browser first, as I was familiar with the name. Unfortunately, it DOES NOT play Flash video. The familiar, "this device does not support flash" message came up.
Then I tried both the free version of the Puffin browser and Diigo Browser, and they both worked exactly as advertised. I experienced halting video on Puffin, but this is so common that it could be due to a temporarily-bad WiFi connection. Beyond that they worked for the job required by the teacher, I didn't explore them in any further detail.
It is nice to see that there are ways to overcome these minor annoyances. (And it is simpler than the version I heard from Brian Brushwood on TWiT: use GoToMyPC from your iPad to login to your main computer and run the Flash-capable browser via that virtual connect.)