Either way, it should be frowned upon as not a nice thing to do, but would it actually be assur?
Yes, it is theft. The creator of the product (in this case the song) has agreed on his terms to provide the product for a set price. Instead of purchasing the song you are consciously resorting to other methods that the creator has not consented to in order to get hold of the product.
The more technical question would be as to what extent you own the song file. When you purchase a physical book it is obvious as to what you've acquired ownership of -- you've bought the physical book so you own the copy from cover to cover. But what about a song file? Does that mean you can upload your file to some site and essentially allow people to "borrow" it? This can be analogous to physical books. Can you put your book copy into a scanner and print more copies? Yes. Can you distribute these copies to others? No.
It seems that for most people the idea of property becomes conceptually blurry when it comes to the digital realm. It is very easy and virtually effortless to just click "copy and paste" and publish something online unlike printing and compiling a book, so it's very easy to not realize that something as simple as that would constitute theft.
The Rambam in Hilchos Geneiva v'Aveida clearly says that someone can't steal the smell of something. Simply, that's because the person you "steal" it from isn't lacking anything! So seemingly that would be the case here, since the original owner isn't losing any money / property just because one more person downloaded it, it's just like smelling something where even though the smell doesn't belong to you, and maybe the owner of the object that smells nice worked really hard on it, but it doesn't damage the owner at all.
Your question needs a short clarification:
Based on the Torah, the Gemmorah differentiates between Gnevah (גניבה), which is a robbery and Gzelah (גזילה) which is any other monetary damage either direct or indirect (e.g גזל שינה - sleep deprivation). The Gemmorah itself started (unfortunately) use those terms interchangeably and that continues through the Halachah till nowadays.
Another point is the very definition of "the property" being stolen. The original, Talmudic Judaism has no idea of a non-physical property, and therefore no idea of stealing such property. Instead, it discusses ways of indirect damages, that can be evaluated in monetary compensation, e.g. blocking someone's view (היזק ראיה) is not considered stealing (גזילה), but damaging (נזק). Therefore, (originally) Judaism does not recognize any idea of intellectual property.
Another important point is the ability to sue. Unlike stealing of the physical property that can be sued or damages that can be compensated, all other types of property cannot be sued. Therefore, as many already pointed, it falls under "general bad stuff" that can only be banned or excommunicated.
"Dina deMalchutah" is the only loophole to allow our Halacha to embrace the idea of the intellectual property, just like any other monetary regulation, I do, however, doubt if any serious Beis Din would discuss such claims.
Yes, downloading "pirated" music is "stealing"*. Next question, please?
If you are asking about specific instances in which someone had permission (explicit or otherwise) to download music from someone who had permission (explicit or otherwise) to upload it, then you're not talking about downloading "pirated" music.
It is a very subtle difference, but it (potentially) makes all the difference in the world.
Ultimately, Dina DeMalchutha Dina. If you have no right in the eyes of the local/state/federal government to download that music because permission needs to be granted to you by someone who has not done so, then it is "stealing"** if you download it. If the person who uploaded it had no right to do so, then it is "pirated". If neither of you had permission to do what you did/are doing then it is "pirated" music and you are "stealing"* it.
*I'm using the word "stealing" here in the way I understand zaq to have used it in the original question - taking something that doesn't belong to you that was provided to the public by someone who didn't have a right to put it there. However, it may not be Halachically stealing; it might be more akin to the secular "receiving stolen property", which might be different. You might have to return it, but you might not face penalties. Or you might NOT have to return it, EVEN THOUGH you might have to pay penalties. That's a separate question. I think the question is about whether or not this is permitted, not about what the technical definition of the activity is.
**I'm using the word "stealing" here in the way I believe is most appropriate. If you are taking something from a domain in which the person who posted it had permission but you do not have permission to download it (eg., a paid, legal subscription service in which you use a hacked membership ID), this would be "stealing" (in some sense of taking something you're not allowed to take) even if it wasn't "pirated".