Magic in Middle-earth

[Nukapei]

I've been thinking (never a good thing! ) about the difference between modern fantasy epics and Lord of the Rings, and one thing that really stuck out in my mind was their different approaches to magic. Whereas most modern fantasies are so chock-full of magic it's hard to breathe, Tolkien is much more subtle in his magic usage. The only things I could think about that were "magical" were the Rings of Power, and the Istari.

Then I got to thinking about Lúthien, and the way she used magic in helping Beren steal the Silmaril from Morgoth, including rescuing him from Sauron. Now I am confused. What role does magic play in Middle-earth? Obviously it can be wielded by semi-divine beings (like the Maia, and the Istari), and contained in an object to be wielded by those who control it (like Lúthien and her suit, and Frodo and the Ring), but what else? Is it the object or the person that has the magic?