A friend, who paid for music files, wants to let me download some tracks on to my MP3 player. Am allowed to do it for a short while, as if I am borrowing them, as long as I do not keep them? Why would it be different than me just borrowing his MP3 player for a short while, to listen to tracks I do not own? 

 

See here.

The same would be in the case you described. Just as you cannot “borrow without permission” for a short while and give it back (aka stealing).

When borrowing the MP3, what was bought you are borrowing. However when making a copy you are “borrowing without permission” something else and the original – bought – item is still usable and in the possession of the buyer and you have a additional unauthorised copy.