Something to keep in mind:
We have to obey local, state, national, and international law unless these laws are unjust. Just laws are established by legitimate authorities, are publicly known to be actual laws, and they do not conflict with divine law.
The law in the United States says it is illegal to make copies of music (this includes downloadig copies) without paying for each individual file (through a legitimate service like iTunes). This law is not unjust, because it has be made by a legitimate authority, it is publicly known that it is illegal to do so (copyrights), and this law does not conflict with Divine law in any way. That in and of itself is the reason we can’t illegally download music (or tape music from the radio, or rip our friends’ CDs/DVDs etc). If this were an unjust law, breaking it would not be enough, we would also have to make an effort to have the law changed.
Now, the “borrowing” argument is defective in this case. If I borrow your car, but never bring it back to you, then I have stolen it. The money that we spend for music typically pays for our right to own a copy of the music and to enjoy it at our leisure. We can lend out our music to others, and while they have it, they are using it in our name. When the library purchases a book, it has paid for the right to own the book and has the right to lend it out. When the user is done, the book comes back. When we lend out a CD, it comes back, but if a copy is made, then the copy is illegally made and is against the law as it was not authorized by the owner of the actual musical recording itseld (the artist).
Now, this might seem unfair. It might seem like a ploy for the record companies to get rich. maybe it is. If so, that isn’t the issue when it comes to morality. Two wrongs do not make a right. So yes, downloading music illegally is a sin. It’s a sin most of us commit in the digital age. I know, I got to confess it to an abbot once, haha, and he agreed that it was stealing.