APNG is designed to be backwards-compatible with image viewers which only support PNG. Older image viewers will simply disply the first frame of the animation, like a normal image. A flag can be set to make APNG-aware viewers ignore the first frame, so that it can be used as a staic representation of the animation.
The image below has 2 frames. The first has "APNG not supported" text, and the second has "APNG is supported" text. A program the does not support APNG will display only the first frame, while an APNG-aware program will display only the second frame.
Each APNG frame has a disposal method specified, which controls how the buffer is handled before drawing the next frame. The buffer can be reset to an empty frame, left unmodified, or restored to the beginning of the previous frame.
"Chompy" is the fun little animated GIF you may have seen on Bugzilla while doing a search. It was made by Klowner (klowner.com / dugnet.com/klown), who graciously provided PNGs from the original rendering so I could assemble an APNG version. You can see that the APNG animation, on the bottom, doesn't have a halo effect on non-white backgrounds. To achieve smooth edges with the GIF, you must pre-blend with a fixed background color.
(The APNG version needs cropped a bit to match the size of the GIF, and I think it also has an extra frame or two compared to the GIF version.)
Like Demo 4, this shows how much better animations can look on alternate backgrounds, compared to a pre-blended GIF. Also note that the drop-shadow at the bottom works everywhere.
(The APNG version looks smoother because it includes more frames, and is also slightly larger in size.)
The "throbber" displayed in the Firefox OS X theme is a pre-blended GIF. This was easy to notice with my Chromatabs extension, which changed the tab color. Aronnax created an APNG version for the GrApple theme, which looks much better.