An interesting topic, nonetheless for having been discussed before. The nature of Tolkien's magic is very different than that of other fantasy books. I already posted something about the nature of magic in another thread so allow me to reproduce it here.
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Magic, I believe, refers always to those things that the speaker does not understand. Therefore the word is used for many different things that are not really related at all. Here I will describe several different forms of "magic" as I see them.
Dwarf: the Dwarves are said to use magic especially when making secret doors. I think that Dwarf magic is nothing more than a type of technology as Mithalwen said earlier.
Ents: the drink that Treebeard gave the hobbits could be said to be magical, but it seems only natural that "tree people" would have such drinks.
Elves: they seem to have great skill in making things: the cloaks, the gems of the Noldor, the Ships of the Teleri, and the Palántiri stones for example. Those Elves who beheld the light of the two trees also had a power within them that repelled evil. Also they could sing songs of power, more on this in a minute.
Tom Bombadil: Tom was one with the land he inhabited, it's life was his. He was master of everything that dwelled in said land and everything he commanded happened.
Valar/Maiar: they shaped the world and their power flowed in every inch of it. Therefore they could, to some extent (depending on their personal power), command the matter of the world. They also sang songs of power.
Music and Magic: Although I can not claim to fully understand how this worked, music had power in Tolkiens world. The foundation of the world was the music of the Ainur, and Sauron had a contest with Finrod using songs of power. And Tom Bombadil used songs of a sort to. There were other examples of course but you get the idea.
This is not a complete list by any means (and you could probably have an entire thread for each type of magic and how they worked) but I was just trying to get the point across that what is referred to as "magic" is not one thing but many different things.
Also, I think that the different kinds of "magic" could be learned by the other races, so that "Dwarf Magic" could be learned by Elves for example. However there are exeptions to this, Tom Bombadil for example. Also there are some things that the Ainur could do that lesser beings could not because they don't have the power required.
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Luthien is a mix between an Elf and a Maia so her powers would reflect that.
Of course there is much more to discuss about magic than what is contained in that rather condensed and simplified version.