What does "hardware acceleration" refer to?
I've seen it used in reference mostly to animations and other visual effects. Fancy CSS3 stuff, flash video playback, things like that. I don't know if I've ever seen it used when it didn't have anything to do with web browsers, though that could just be a result of me not reading any technical literature that doesn't have to do with web development.
Answer
It's when you have hardware designed to accelerate a specific function.
So, for example, when you decode or encode video you do it on the CPU. Some video cards allow you to do this on their GPU instead, so you now have "hardware accelerated video decoding".
So two users, one with hardware acceleration and one without will see more or less the same visuals, but the one without may experience less frames or resolution etc. Or it might just mean a different part of the computer gets hotter instead and it looks the same on both.
Another example, you can use a 3D package and you might get 30 fps. If you add a posh graphics card that the 3D package can use directly you can now get 60 fps. The hardware has accelerated the process!