I've seen this in many videos, especially music videos.
I have some idea of how this works, but can anyone tell me the best way to create the effect where one object or person is standing still and the background is moving fast (or very slow) behind the main subject?
These days, it's generally a composite, the front layer is the talent, the back layer or "plate" shot separately, using a motion control rig that can exactly duplicate it's moves over multiple takes.
The older method does everything "in the camera", i.e. in one layer. To do that, you speed up the audio of the song that is played on the set for lip-synch, shooting at your highest frame rate, the singer sings or lip-synch's faster to match it, then when you slow down the footage in post, the singer seems to be in normal speed while the background seems slowed-down.