Three kinds of magic in stories

When magic occurs in stories it has various flavours: here are three of them. 

One is the low magic of "sword and sorcery" stories, and indeed most stories that feature magic - in which magic functions as a technology of power. This ranges widely across spells, and conjurations, summoning of powerful beings, seeing the future, controlling others etc. 

But this is not a very "magical" use of magic - indeed it reduces the magical to the mundane. 

Indeed, such stories are often cynical, reductionist, intended to dis-enchant. 

There are also stories in which magical elements function to indicate a hidden world - the magic enables perception of what would otherwise be imperceptible or hidden: this is the "occult revealing" function of magic. 

Such magic may reveal a world of spirits behind the material world, that "the dead" are alive and can be communicated with, that there are invisible manifestations (such as auras), that telepathy is real - and such like. 

The general impression is that there is more to life than commonly acknowledged, and this can impart an enchanted atmosphere.

The third kind of magic is pure enchantment - and this is (pretty much) how the magic of elves is depicted in The Lord of the Rings. 

There is very little use of magic as a technology - although there is some (the Mirror of Galadriel; and elven swords, cloaks, and ropes for instance). There is also the occult trope, in the hobbits often comment that the world turns-out to contains much more and strange phenomena and entities than the commonplace experiences of the Shire would suggest.   

But the main use of elf magic is related to an atmosphere of enchantment; a different quality of attitude, experience, motivation among elves; a feeling that is both joyous and sad, supremely good and perilous - here-and-now and yet backward looking across vast expanses of time.

 As implied - these uses of magic have somewhat different narrative functions in stories, and presumably different intent. When magic is used as a technology, there may be an element of intending to induce wishful thinking in the reader; but on the whole the magic is just a narrative device, among many. 

When magic is used to indicate occult realities, this may have the intent of indirectly encouraging the reader to take the same attitude to his ordinary life - the story may be encouraging the reader to assume that behind the mundane world is another reality... if only a different attitude were adopted to perceiving it, or particular abilities were available.   

The purpose of pure enchantment in a story is - I take it - religious - in the sense of a whole-world view.

The magic of enchantment in a story carries - to some degree - the implication that "reality" - our life, nature - really is a more wonder-full and inspiring thing than we generally assume.