What exactly is it? When would you use it? Some training tips on correctly using it.
It is more commonly called "forearm rotation" because the forearm is what does the work, not the hand. The basic concept is that instead of doing all the work of playing by raising and lowering your fingers with your hand still, you rotate your forearm "inwards" (so your thumb moves down as your hand rotates) to help play a note with your thumb, and the opposite way to play a note with your little finger.
This assists strength and stamina, since the big muscles in your arm are more powerful than the smaller muscles that move your fingers.
The current "guru" of this technique seems to be Dorothy Taubman, but it goes back much further. My teacher introduced me to it from books on piano technique by Tobias Matthay — but that was long before internet videos were available
I have actually thought about this some more and am not sure I fully agree with equating forearm rotation with hand rotation as suggested by the videos that were quoted. I should clarify that I am speaking here from my own intuition and not from strict definitions.
I think the hand rotation can certainly be initiated from the forearm as very well explained in the videos, but the wrist can also initiate it, and typically a full rotation has to be a combination of both.
The inside-outside oscillations initiated by the forearm muscles--the video has a more technical term for them--leads to the hand oscillation between left and right weights, the only thing you need for a tremolo 1-3-1-3 or 1-5-1-5 for example.
But for an up and down movement alone, that's something the wrist is best at initiating just from a mechanical standpoint if nothing else.
To me, a full hand rotation movement is accomplished by the combined action of the forearm and wrist. At least this is how I visualize it when playing pieces like Chopin etudes 1 or 2 op 25 just to name two very visible examples of rotations, as opposed to say etude 11 op 25 where there you have more of an oscillation movement instead of a rotation.