This is when you have hardware designed to speed up a specific function.
So, for example, when decoding or encoding a video, you do this on the CPU. Some video cards allow you to do this on your GPU, so now you have "hardware accelerated video decoding."
Thus, two users, with and without hardware acceleration, will see more or less the same visual effects, but one without it may have fewer frames or resolution, etc. Or it could mean that the other part of the computer is getting hotter and it looks the same on both.
In another example, you can use a 3D package and you can get 30 frames per second. If you add a smart graphic card that the 3D package can use directly, now you can get 60 frames per second. Hardware accelerated the process!