Magic in Middle-earth
It has often been remarked that there isn't a lot of magic in Middle-earth. Compared to fantasies that have come after it - and, in particular, those that have tried to imitate it - The Lord of the Rings exists in a pretty mundane world.
Even though some Shire residents call Gandalf a conjurer, he never "conjures up" anything or anybody. Dwarves can do marvelous things with their hands, but this is through skill, not magic. Ents are strange and strong, and their drink is marvelously invigorating, but even they don't perform what we would usually call magic. Mortal races (Men and Hobbits) don't use it at all. And Elves... Well, we'll talk about Elves later.
But, yet, Middle-earth is a magical place. Where else could the rings of power exist, or the Old Forest, or Lothlórien, or the Grey Havens that lead to the straight path to Valinor? Where is all that magic hiding? Before we start hunting, it may be useful to look at what "magic" is in Tolkien's secondary creation.
In Middle-earth, the more closely associated someone is with magic (i.e., what mortals would think of as magic), the harder it is for that person to define it. Shire residents are quite clear on the subject - and stay as far away from the stuff as possible. Compare this to Galadriel's comment that her mirror is what Hobbits would call magic, "...though I do not understand clearly what they mean; and they seem to use the same word of the deceits of the Enemy." To those inhabitants who understand it, one thing magic in Middle-earth is not is neutral.
That's because magic in Tolkien's creation doesn't come ultimately from the person who is wielding it. It's not a skill that someone can learn and then practice. "...it is not to be come by by 'lore' or spells; but is an inherent power not possessed or attainable by Men as such." (Letter #155) Magic is power, and that power has to come from somewhere. When Gandalf faces the Balrog, he doesn't call on his own abilities as a wizard; he says, "I am a servant of the Secret Fire, wielder of the flame of Anor. You cannot pass." He acknowledges that the Balrog also gets its power from elsewhere: "The dark fire will not avail you, flame of Udûn. Go back to the Shadow!" The question in this contest is not whether Gandalf or the Balrog is stronger; the question is which of them serves the more powerful Fire. (We don't learn anything more about the Secret Fire in LotR, but in The Silmarillion it is said to be "with Eru" and in at least one letter Tolkien identified it as the Holy Spirit.)
Saruman is able to be overthrown only after he has deserted the Power he originally served, deciding to become his own master. He has broken the power he had been entrusted with into its constituent parts, and thinks this means he understands it and can wield it through his own abilities. However, he succeeds only in making himself vulnerable to a lesser Power than the first. Gandalf, when he is sent back to Middle-earth after he passes through death, is given Saruman's original unbroken power and becomes "Saruman as he should have been."
Note that Gandalf is sent back to Middle-earth - he doesn't come back of his own accord. When the work he has been sent to do is completed, he goes home. At the death of Saruman, and also at the downfall of Sauron, we see the fate of those who trusted in themselves; their pitiful attempts to return home through their own power are pulled apart by the wind. Gandalf, Saruman, and Sauron are Maiar (angelic beings, compared to the Valar who might be called archangelic beings), so their differing departures from Middle-earth can be legitimately compared. The only one who retains the power he has been given is Gandalf, who has neither claimed it as "his" nor tried to increase it for his own purposes.
So, where does the power lie? Who is doing the sending? The hierarchy extends through the Valar to Eru Himself. Most of the comments I've read on the subject by Tolkien scholars agree that, although the Valar may have originally sent the wizards to Middle-earth, they could not have sent Gandalf back after his death; this must have been done directly by Eru, as the power over life and death in Tolkien's cosmos belongs to Him alone.
To Eru's children (Elves, Men and, by extrapolation, Hobbits) he is also Ilúvatar, or Father. He has set out different paths for them. Mortal peoples will die, and pass from the created world to a fate that lies beyond it. The Elves, however, are bound to the created world (which includes Valinor) in a deeper way, because that is where their fate lies until the end of time. Not even Elves who are killed or die of grief leave the created world; they are gathered into the Halls of Mandos, one of the Valar. The power of this bond to the created world, a power given the Elves in their very nature by Eru, is shown in what Sam calls "elf-magic."
Mortals do not have this bond, so do not have the Elves' innate ability to affect and be attuned to the natural world. However, the Elves call mortality itself "the Gift of Ilúvatar." This can be partially explained by the fact that an Elf can eventually tire of life and would then see death as a gift, but there is more to it than that. As Elves are bonded to the created world, so mortals are bonded to something beyond it - something the Elves do not reach until the end of all things. In the first chapter of The Silmarillion, we learn that mortals have the ability, through their freely chosen actions, to actually alter the Song that lays out the path for all of creation; Elves do not, although they have the choice of how to respond to it. Therefore, we know that even while mortals live within the created world, their bond to something beyond it already exists.
Mortals don't call this bond "magic," as, to them, it's part of the choices and actions that make up their normal daily lives. In the same way, the Elves' bond to the natural world is completely normal to them, so they're unclear (as Galadriel says) as to what mortals mean by "elf-magic."
In order to save Middle-earth in LotR, the free peoples call on all three kinds of "magic": the direct gift of Power from the Valar and Eru held by Gandalf, the abilities inherent in the Elves' bond with the natural world, and the ability of mortals to freely make choices and act on them, and by so doing to participate in determining the outcome of the Song.
None of these gifts and abilities, however, are magic as we might normally define it. They are natural abilities and supernatural powers, all given by Eru to his creatures according to their kind. Middle-earth seems mundane because it is: that is, it's as mundane as is our primary creation as Tolkien understood it, which is filled with natural abilities and supernatural powers of its own. There's a sense of loss when the Elves leave Middle-earth at the end of the Third Age; their "natural magic" is no longer part of the world of Men. But the supernatural nature of Men holds the stronger power. After Gandalf's departure, there are no angelic beings living bodily in Middle-earth. But that doesn't mean Men have any less supernatural assistance available to them; it just means that, perhaps, they don't have the same need to see it physically.