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Magic was a term used by the Hobbits to describe processes and abilities of objects which could not be explained out of their limited lore and knowledge.
- "For this is what your folk would call magic. I believe; though I do not understand clearly what they mean; and they seem also to use the same word of the deceits of the Enemy"
- ― The Fellowship of the Ring, The Mirror of Galadriel
"Magic" for Elves and Wizards, was not something special or different from the natural world, just part of it that was not immediately visible, leading to more common categorization of things as "Seen" or "Unseen". The "Seen" aspects of a thing or person are only part of it. Through the power of the One Ring, Frodo Baggins at times saw powerful Elves, such as the High-elf Glorfindel, in their true level of power and radiance that extends into the Unseen.
The effects which could be described magical, were of different natures.
The Maiar as supernatural beings could alter the world in terms according to their nature. Sauron being a Maia used his powers to manipulate his slaves and objects. Some of them he taught to the Elves who created the Rings of Power, as well as his slaves, the Black Númenóreans and the Kings who became Nazgûl, who used this lore as "sorcery".
Similar magic was the one used by the Istari, the Maiar who came to Middle-earth to help the Free Peoples against Sauron. The nature of the Istari was not known to all, and they were known as wise old men. They were called Wizards.
Properly, Men using "magic", even for good, are not termed "Wizards". In Tolkien's mythology, this term is specifically only used for the Order of the Istari, who are of angelic origin. A member of the race of Men wielding magical powers would be referred to as a sorcerer, whether for good or evil.
The Elves were not supernatural beings and although knew the powers of the Ainur, they did not seem to comprehend the concept of "magic". The Elves were taught arts and crafts by the Valar in Valinor and could put dedication and love in every thing they wrought. Due to their immortality, Elven smiths could reach high levels of experience in craft and lore.
As mentioned, the Rings of Power were such "magical" objects that would seem supernatural to the regular people; they gave powers of manipulation of the word and invisibility. Other Elven artifacts were the Palantíri or the Lamps of the Noldor, as well as the Mirror of Galadriel. Some simpler artifacts were the river-boats of the Galadhrim and the elven ropes, which seemd to have will on their own.
Tolkien uses the word "Sorcery" and "Sorcerer" in negative context. Sorcery is the equivalent of the dark magic. Sauron taught by Sauron to some of his followers like the Witch-King of Angmar and the Mouth of Sauron and perhaps other Black Numenoreans.