There is much magic in Middle Earth, but there is not a category of mortal magicians. There are a number of categories of beings with inherent magic. There are the wizards (i.e. the five Istari), and there are the ring-bearers (and there is Gandalf who was both). There are also races with power, such as Elves and Ents. There are also figures like Tom Bombadil.
There can be mortals involved with the supernatural, but they shouldn't be viewed as "magicians" -- i.e. someone who does magic as the main thing they do, or who works magic as a profession. Mortals can talents like turning into a bear, or speaking to thrushes, or building items of great power -- such as the impregnable tower of Orthanc, or the barrow-blade that killed the Witch-king, or the secret door into the Lonely Mountain that opened once a year. As GM, I am comfortable with extrapolating to magic which is not canonically found in Tolkien. However, I want to avoid importing magic concepts from other fantasy works or games -- especially Dungeons & Dragons.
Magic in Middle Earth is based around enhancing or weakening the inherent properties of things. It draws from the inherent power of both the practitioner and the object of the magic. Thus, effects which are truly unnatural are very limited.
All magic has definite limitations, however.
"If Gandalf would go before us with a bright flame, he might melt a path for you," said Legolas. The storm had troubled him little, and he alone of the Company remained still light of heart.
"If Elves could fly over mountains, they might fetch the Sun to save us," answered Gandalf. "But I must have something to work on. I cannot burn snow."
The Fellowship of the Ring (284)
The magic of the Maiar -- such as the five wizards and Sauron -- is essentially unique. They may imbue another with power, such as The Mouth of Sauron. There are also "black arts" mentioned which Men practiced -- though it is not clear whether there were also "white arts" or what precisely the black arts were. For the Maiar, there are some definite effects known:
"Now go on again!" said Beorn to the wizard.
"Where was I? O yes -- I was not grabbed. I killed a goblin or two with a flash --"
"Good!" growled Beorn, "It is some good being a wizard then."
Gandalf & Beorn, The Hobbit
"I will come," said Gimli. "I wish to see him and learn if he really looks like you."
"And how will you learn that, Master Dwarf?" said Gandalf.
"Saruman could look like me in your eyes, if it suited his purpose with you. And are you yet wise enough to detect all his counterfeits?"
The Two Towers (562)
Legolas gave a great shout and shot an arrow high into the air: it vanished in a flash of flame.
"Mithrandir!" he cried. "Mithrandir!"
"Well met, I say to you again, Legolas!" said the old man. . . .
At last Aragorn stirred. "Gandalf!" he said. "Beyond all hope you return to us in our need! What veil was over my sight? Gandalf!"
The Two Towers (483 - 4)
"There's more behind this than sun and warm air," [Sam] muttered to himself. "I don't like this great big tree. I don't trust it. Hark at it singing about sleep now! This won't do at all!"
The Fellowship of the Ring (114-5)
"I could think of nothing to do but put a shutting-spell on the door. I know many; but to do things of that kind rightly requires time, and even then the door can be broken by strength."
Gandalf, The Fellowship of the Ring (318-9)
Once he (Saruman) was as great as his fame made him. His knowledge was deep, his thought was subtle, and his hands marvellously skilled; and he had a power over the minds of others. The wise he could persuade, and the smaller folk he could daunt. That power he certainly still keeps. There are not many in Middle-earth that I should say were safe, if they were left alone to talk with him, even now when he has suffered a defeat. Gandalf, Elrond, and Galadriel, perhaps, now that his wickedness has been laid bare, but very few others.
LOTR, Book III, chapter 10
Gandalf struck a blue light on the end of his staff, and in its firework glare the poor hobbit could be seen kneeling on the hearth-rug, shaking like a jelly that was melting.
The Hobbit (27)
'Knock on the doors with your head, Peregrin Took', said Gandalf. 'But if that does not shatter them, and I am allowed a little peace from foolish questions, I will seek for the opening words.
'I once knew every spell in all the tongues of Elves or Men or Orcs, that was ever used for such a purpose. I can still remember ten score of them without searching in my mind.'
The Fellowship of the Ring, Book I, page 400
The Men of Númenor were settled far and wide on the shores and seaward regions of the Great Lands, but for the most part they fell into evils and follies. Many became enamoured of the Darkness and the black arts; some were given over wholly to idleness and ease, and some fought among themselves, until they were conquered in their weakness by the wild men.
It is not said that evil arts were ever practised in Gondor, or that the Nameless One was ever named in honour there; and the old wisdom and beauty brought out of the West remained long in the realm of the sons of Elendil the Fair, and they linger there still. Yet even so it was Gondor that brought about its own decay, falling by degrees into dotage, and thinking that the Enemy was asleep, who was only banished not destroyed.
LOTR, Book IV, chapter 5, subpage 20.
The rider was robed all in black, and black was his lofty helm; yet this was no Ringwraith but a living man. The Lieutenant of the Tower of Barad-dur he was, and his name is remembered in no tale; for he himself had forgotten it, and he said: 'I am the Mouth of Sauron.' But it is told that he was a renegade, who came of the race of those that are named the Black Nümenörians; for they established their dwellings in Middle-earth during the years of Sauron's domination, and they worshipped him, being enamoured of evil knowledge. And he entered the service of the Dark Tower when it first rose again, and because of his cunning he grew ever higher in the Lord's favour; and he learned great sorcery, and knew much of the mind of Sauron; and he was more cruel than any orc.
LOTR, Book V, page 202
In addition, the Valar and the Maiar have the ability to change their bodies. Any Valar or Maiar can body or disembody themselves at will. Thus, killing their body does not kill them, not for a long time. Nor can you be sure their shape will ever be the same twice. Yet, somehow, Middle-Earth itself wears away this capacity, even in good Maiar like Radagast and Gandalf.
Now the Valar took to themselves shape and hue; and because they were drawn into the World by love of the Children of Ilvatar, for whom they hoped, they took shape after that manner which they beheld in the Vision of Iluvatar, save only in majesty and splendour. Moreover their shape comes of their knowledge of the visible World, rather than of the World itself; and they need it not, save only as we use raiment, and yet we may be naked and suffer no loss of our being.
The Silmarillion (11)
There are some subtle effects whose power is unclear. There are cases of characters seeing the future (or possible futures), and giving blessings or curses on others.
He laid his hand on the pony's head, and spoke in a low voice. "Go with words of guard and guiding on you," he said. "You are a wise beast, and have learned much in Rivendell. Make your ways to places where you can find grass, and so come in time to Elrond's house, or wherever you wish to go."
Gandalf, The Fellowhip of the Ring (295-6)
"Thus we meet again, though all the hosts of Mordor lay between us," said Aragorn. "Did I not say so at the Hornburg?"
"So you spoke," said Eomer, "but hope oft decieves, and I knew not then that you were a man foresighted."
The Return of the King (830)
The Elves clearly had great powers of crafting magical items.
Boromir leaped forward and hewed at the arm with all his might; but his sword rang, glanced aside, and fell from his shaken hand. The blade was notched.
Suddenly, and to his own surprise, Frodo felt a hot wrath blaze up in his heart. "The Shire!" he cried, and springing beside Boromir, he stooped, and stabbed with Sting at the hideous foot. There was a bellow, and the foot jerked back, nearly wrenching Sting from Frodo's arm. Black drops dripped from the blad and smoked on the floor. Boromir hurled himself against the door and slammed it again.
"One for the Shire!" cried Aragorn. "The hobbit's bite is deep! You have a good blade, Frodo son of Drogo!"
The Fellowship of the Ring, Book II, chapter 5, subpage 6.
Many of the Ents were hurling themselves against the Orthanc-rock; but it defeated them. It is very smooth and hard. Some wizardry is in it, perhaps, older and stronger than Saruman's. Anyway they could not get a grip on it, or make a crack in it; and they were bruising and wounding themselves against it.
LOTR, Book III, chapter 9, subpage 11.
So passed the sword of the Barrow-downs, work of Westernesse. But glad would he have been to know its fate who wrought it slowly long ago in the North-kingdom when the Dunedain were young, and chief among their foes was the dread realm of Angmar and its sorcerer king. No other blade, not though mightier hands had wielded it, would have dealt that foe a wound so bitter, cleaving the undead flesh, breaking the spell that knit his unseen sinews to his will.
LOTR, Book V, chapter 6, page 6.
There are many feats of magic associated with interacting with animals and plants. Bard of Laketown found that could speak with the old thrush, for example -- which was his birthright from the race of Dale. Beorn could clearly talk to beasts as well, and there were birds which acted as spies for Saruman. Beorn could change into a bear as well -- he was human, not a Maiar, but it is not clear where his power came from.
Suddenly out of the darkness something fluttered to his shoulder. He started -- but it was only an old thrush. Unafraid it perched by his shoulder and it brought him news. Marvelling he found he could understand its tongue, for he was of the race of Dale.
The Hobbit (236)
. . . a whole regiment of birds had broken away suddenly from the main host, and came, flying low, straight towards the ridge. Sam thought they were a kind of crow of large size. As they passed overhead, on so dense a throng that their shadow followed them darkly over the land below, one harsh croak was heard.
The Fellowship of the Ring (277-8)
"He is a skin-changer. He changes his skin: sometimes he is a huge black bear, sometimes he is a great strong black-haired man with huge arms and a great beard. I cannot tell you much more, though that ought to be enough."
Gandalf, The Hobbit (116)
It is worth noting that besides genuine magic of the sort that Gandalf used, there were lesser magic which may or may not have had any real power. For example, the dwarves in The Hobbit cast "spells" of some sort to hide their troll-cache.
Then they brought up their ponies, and carried away the pots of gold, and buried them very secretly not far from the track by the river, putting a great many spells over them, just in case they ever had the chance to come back and recover them.
The Hobbit (51)
The most powerful beings inhabiting Arda are the Valar and the Maiar -- the original Ainur who have created the world directed by Ilúvatar. They have governed the world ever since, and fight against the Shadow. Unlike the traditional mythological deities, they are real beings, which, however, today inhabit dimensions unreachable by physical means. They do not interfere directly in the business of Arda's inhabitants, although a long time ago, one could even meet them and talk to them (however, the privilege was reserved for carefully selected individuals, such as Tuor or Eärendil).
The Valar, the most powerful rulers of the world, decided to retreat from Arda, transporting Aman -- the continent they lived on -- to a special, ensorcelled dimension, where only the ones allowed to enter have access to. The majority of the Maiar have stayed there with them, although many of them continued to visit Middle-earth -- especially their "offspring", who were born during the First and the Second Age (or perhaps even during the time of Creation before the appearance of the Two Trees), and who became fond of Middle-earth as their homeland, often bonding themselves to a certain material object of nature: a tree, a rock, a lake, a mountain etc. The examples are those of the spirits of Melian, Caradhras, the Silent Watchers guarding the entrance to Minas Morgul, Tom Bombadil and Goldberry etc. Nevertheless, they, too, mostly hide from mortals, using spells like Sanctuary or Plane Shift to escape in case someone spots them.
The only Maiar to openly arrive to Middle-earth during the Third Age were the Istari, five wizards, most notable being Gandalf and Saruman. By arriving to Middle-earth their souls have closely bonded with their physical forms, even to the extent of Saruman's physical form (together with its needs: greed, the wish for more and more power etc.) prevailing over his soul in the end.
Some of the Maiar fell under the influence of Morgoth before arriving to Arda, and many of them stayed active long after their master's fall: except for Sauron, the most famous of them would be the Balrog, and there is a lot of lesser, weaker demons hiding themselves under the earth and in other planes of existence, wherefrom they can be summoned by the evil or the careless wielders of magic.
As Eru's "first-born", the Elves are the closest to magic of all the free peoples of Middle-Earth. Since the first time their leaders were brought to Valinor, they have been studying magical powers from the Valar and the Maiar, mostly those concerned with artifact creation (the Noldor) or the coexistence with nature (the Sindar).
Created by Aulë, the Dwarves are not the true children of Eru, although they are also considered one of the Free Peoples. Therefore, their connection with magic is the weakest, and they weren't even aware of magic's existence until they met the Elves who gave them runes. Together with runes, they also learned the skill of using runes to hone the products of their crafts, and ever since then this has been the only form of magic known to the Dwarves.
Men, as the second-born of Ilúvatar, are somewhere between the Elves and the Dwarves when it comes to their relation with magic. Just like Elves, Men can learn all the forms of magic use, however, they do not have any inborn magical abilities.
In the early days of ancient Númenor, under the patronage of Elves who visited the island frequently, Men have built their new realm with an abundant supply of magical help. Judging by the quantity of spells being learnt and used, and the magical items that survived the Fall, or perished and transformed to legends, Númenor was probably a high-mana area. Unfortunately, many mages fell under the influence of Sauron when Ar-Pharazôn brought him to the island, so they perished with Númenor, together with their knowledge and skills. Some items were left behind, and came to Middle-earth with the Dúnedain or the Black Númenoreans. The ancient Númenoreans have also created cities and buildings, such as the fortress of Ostiras (later Hornburg) or the haven of Umbar. The ones who survived the Fall by seeking shelter on Middle-earth have discovered that their powers were much weaker here, and that they wear out a lot more frequently. Together with the weakening of the Line of Númenor and the weakening of their magical talents, it has caused a gradual weakening and forgetting of their magical skills. Nevertheless, during the first years of their power, the Dúnedain have built glorious citadels like Isengard, Minas Tirith and the towers of Osgiliath, while the ones under the influence of the Shadow -- the Corsairs and the Black Númenoreans -- continued using the magical traditions that have always been considered evil.
Concerning the other cultures of Men, most of them have met some forms of magic in their past -- mostly from the Elves or the Númenoreans, and later from the five Wizards. During such experiences, most of the cultures have developed a negative, superstitious disposition towards magic, especially the ones under the influence of the Shadow, while the others learned some skills by themselves. Some of the Peoples, whose ancestors had met some of the Valar and learned a correct attitude towards nature in the ancient times, have developed a tradition -- although different from nation to nation -- called Animism.
An example of dark magic of men would be the Lieutenant of Barad-dûr, the nameless Mouth of Sauron. No more is said of him in the draft of the history than that "It is told that he was a living man, who being-captured as a youth became a servant of the Dark Tower, and because of his cunning grew high in the Lord's favour ..." In the fair copy this was repeated, but was changed subsequently to: "But it is said that he was a renegade, son of a house of wise and noble men in Gondor, who becoming enamoured of evil knowledge entered the service of the Dark Tower, and because of his cunning [and the fertile cruelty of his mind] [and servility] he grew ever higher in the Lord's favour ..." (these phrases being thus bracketed in the original). In RK (p. 164) the Mouth of Sauron "came of the race of those that are named the Black Numenoreans."
Other than that we really know little of him except; he could ride a horse; he was a powerful sorcerer; and he is a trusted emissary for Sauron. There is question as to his age, the Silmarillion gives us some clues on the nature of certain Black Numenoreans.
'Now this yearning grew ever greater with the years; and the Numenoreans began to hunger for the undying city tht they saw from afar, and the desire of everlasting life, to escape from death and the ending of delight, grew strong upon them; and ever as their power and glory grew greater their unquiet increased... But the fear of death grew ever dearker upon them, and they delayed it by all means that they could; and they began to build great houses for thier dead, while thier wise men laboured unceasingly to discover if they might the secret of recalling life, or at the least of the prolonging of Men's days.'
We know that he is old, entering the service of Sauron in 2951, and it can discerned from THE BLACK GATES OPEN that he had magically extended his lifespan, forgetting unimportant details like his name in the process. Several of the Black Kings had age spans equal to or greater than Aragorn. I tend to think as being older than Aragorn, well over 100 years old at the time of the War of the Ring. His bloodline, sorcery, and Saurons aid would all have contributed to his long life span. One could easily assume a pure-blood Black Numenorean could live to be 180 - 300 years.
As all the other creatures (except for the Elves), the Orcs have no inborn magical abilities. However, the rare among them deserving the trust of their masters (mostly Sauron, Saruman and the Nazgûl) may have been given access to some simpler effects. For example, the orcs who captured Merry and Pippin had a healing draught.
The following links are for inspiration, but I am not adhering to the view of magic of any particular one of them.