When I'm helping new InDesign users understand exactly what "flattening transparency" means, I like to show them a concrete example. One way is to export the same simple InDesign file (containing a few overlapping transparent objects) to PDF twice: Once as an Acrobat 4-compatible (PDF 1.3) PDF, which flattens transparent objects, often splitting them up into "atomic regions;" and again as an Acrobat 5-compatible (PDF 1.4) PDF, which supports transparency and thus doesn't flatten.
With Acrobat Pro's Edit Object tool (part of its Advanced Editing arsenal), you can drag individual atomic regions of a flattened image around on the page to show how it was split up. In contrast, dragging objects around on an un-flattened PDF is virtually the same as dragging them around the InDesign file itself, because the overlapping transparent objects remain intact.
Another InDesign trainer on the ID listserv told me of an even better way to demonstrate this, and I thought anyone using InDesign might like to try it for themselves.
Adobe Illustrator, since version 9, has had the ability to create transparent objects and flatten them right in the Illustrator file (something you can't do in InDesign CS). You can move the split-up objects around with the Direct Selection tool to see the results, and you can compare them to an unflattened duplicate on the same page.
To use the live flattener, select overlapping objects in the Illustrator file with your Selection tool. Then open the Flatten Transparency… dialog from the Object menu, choose a Flattener resolution setting (in Illy CS, they're the same as the ones in InDesign CS) and click OK.
To see how the selection had to be split up into a combination of raster and image objects, drag them around the page with the Direct Selection tool. Tres cool!