Captain Code is exactly right, given the context. The Mac OS has always ASSUMED 72dpi. In other words, 1 point = 1 pixel. This is in spite of the fact that modern displays have a physical resolution of more like 80-100dpi. What does this mean? It means that on-screen elements will have a different physical size depending on the the display. In other words, their physical size is dependent on the resolution of the display. For example, what happens when you change your monitor's resolution from 1280x1024 to 640x512? All the elements look twice as big. Why? Because the OS still thinks "1 point = 1 pixel", and since each pixel is now huge, so are all the elements.
With resolution independence, it could instead make every element use half as many pixels, so it would have the same physical size. Or at least, that is one way to use such functionality. That's really not the primary appeal (IMHO), but it demonstrates the concept.
I'm anxiously awaiting more info of Leopard's implementation. There a million ways Apple could offer "resolution independence". I'm hoping you will be allowed to customize this on an app-by-app, or even window-by-window basis. When Tiger was first released, it had a primitive form of resolution independence (not accessible to normal users) which you could set on an app-by-app basis with some debugging tools, so I think that is in fact what they have in mind.
Edit: Actually, saying the Mac OS has "always assumed 72dpi" is not entirely true. The Quartz graphics system, which has been around since 10.0, is largely resolution-independent. But this capability is not used at the system level (and it's probably not sufficient for use at the system level).