Why make an object to contain power of any sort?
If one person does not have power or wants present power increased, he/she can add on to his strength with the object. If he wants to hide his power to appear less dangerous, he can place it elsewhere to re-attain later. For instance, Sauron placed much of his power in the One Ring. Also, a person who carries such power could pass the strength on to another who can put it to better use, as Cirdan gave the ring of fire to Gandalf as he would have greater need of it. Also, some people just enjoy making amazing, beautiful things, perhaps attuned to the singing of Aule.
Why put power into rings as opposed to other objects?
If he does not want the power to be lost at his death or given to one he considers unworthy, he can hand it over to a person of his choosing. Bilbo passed the One Ring on to his adopted heir, Frodo.
Speculation: Rings are small, beautiful, may appear to be an ordinary item of clothing, are easy to show off or to hide, and are easy to pass on to someone else physically and secretly. A staff is obvious so that some doorward like Hama may ask for it to be left at the gate unless talked into letting the person keep it.
Who made the Rings of Power?
In Eregion, which human men called Hollin, lived Noldor elves, craftsmen of the Gwaith-i-Mirdain, the People of the Jewel-smiths. They were unusual among elves in their interest in crafts, and held a cautious friendship with the dwarves of like mind who lived not far away in what the elves called Hadhodrond, and the dwarves first called Khazad-dum and later called Moria. Only Feanor, creator of the Silmarils, had ever shown greater skill among the Children of Iluvatar. Celebrimbor, son of Curufin, was greatest in skill of these Jewel-smiths. Celebrimbor eschewed the evil deeds of his father, remaining in Nargothrond when his father and Celegorm were exiled.
Times seemed peaceful, with Melkor (called Morgoth by Feanor after the proof of his evil) overthrown some years back, and the Valar allowing the people to live their own lives. The Children of Iluvatar felt safe. The elves who stayed behind in their beloved Middle-earth did, however, feel uneasy that they had not heeded the call of the Valar as others had and gone on to Valinor.
However, Melkor left Sauron with his mind still bound by evil words and having done evil deeds. Sauron felt that the Valar had forgotten Middle-earth, and that his offer to repent had been stopped by the fear of humiliation if he had to face the judgement of Manwe as Eonwe had suggested. So he used his ability to assume a fair, noble, wise form, his knowledge of people's desires, his remembrance of his time as a Maiar working with Aule in Valinor, and his fair words to sway others into letting his will be theirs. He stayed away from Lindon, where Elrond and Gil-galad doubted him, still leaving him much room to work. He took on another name, Annatar, the Lord of Gifts.
Annatar motivated the Jewel-smiths into making the rings of power, saying that together they could make the dark, desolate land fair as Eressea or even Valinor, and to raise the scattered, less-civilized elves up along with them to the greatness of power and knowlege of those in Valinor. The Jewel-smiths gladly learned to improve their skill and subtlety and yet also gain some or even much of the bliss they feared they had lost by not entering Valinor. Annatar's great knowledge did indeed improve their abilities, even as he had said, and their works improved vastly, until they took thought and made Rings of Power. Annatar/Sauron knew all of this and guided them, planning to bond the elves to him and to be able to always see what they did.
The elves made many rings at this time, having magical abilities. The power was very great, the three greatest being made by Celebrimbor, the best of the Jewel-smiths. Therefore, when Annatar created a One Ring to bind all of the other rings to it and thus to him, it had to have immense power. The only way to do this was to pass much of his own strength and will into the One Ring, the Ruling Ring, the Great Ring. This act took much daring. While wearing it, he was aware of all that was done with the other rings. He could even see and govern the thoughts of those wearing those rings. The One Ring, having so much of Sauron in it, would try to return to him so that they could both be whole.
What happened to the Rings of Power?
But as soon as Annatar placed the One Ring on his finger, the elves became aware of him and his designs. They took off their rings, and he considered himself betrayed. In fury, he started an open war demanding all the rings for himself, since the elves could not have made them without his teaching. The elves fled, saving the three greatest of the Elven-rings (Narya, Nenya, and Vilya) and hiding them with the Wise. Sauron wanted these rings the most, as they warded off the decay of time on the user and the land. In the war, Eregion was destroyed, Celebrimbor slain, and Moria's doors shut.
The Wise never used the rings openly as long as Sauron wore the Ruling Ring, so the Rings were never defiled by Sauron who never touched them, so he did not win the object of his war. At this time, Elrond Half-elven, one of the Wise, founded Imladris, also called Rivendell and the Last Homely House, his Ring safe-guarding the place.
Unfortunately, Sauron collected all the other Rings of Power, giving them out as he saw fit. He gave the Seven to the Dwarves, and the Nine to Men. He perverted the rings so that they betrayed their users in the end.
Sauron's giving of the Seven Rings may have been allied to the idea of the Seven Fathers of the Dwarves, the original seven created by Aule whom Eru endued with life, who started the Seven Houses. But the Dwarves proved hard to dominate, and they could not be turned to shadows. They used their rings only to gain wealth, but this caused greed for gold and wrath to rise up in them, creating terrible mischief. The Seven Hoards of the Dwarves were apparently created with the Rings, the hoards eventually falling to the ownership of dragons. Some of the rings were destroyed by fire, the others eventually regained by Sauron.
The Men who used their gifts of the Nine Rings became mighty warriors, sorcerers, or kings. They could be invisible to normal men, yet see things of other worlds. Their lives lengthened tremendously, but those lives became unendurable, their bodies becaming ghost-like wraiths in the realm of shadows, slowed in the transformation if their original intentions had been good and their strength high. They were called the Ulairi, and named by Sauron the Nazgul, the Ringwraiths. They were the most fearsome of Sauron's servants, able to send out a sickening darkness some called the Black Breath, and their voices could cause numbing fear.
Sauron became a king and god to the men outside of the elf-lands of Lindon and the Grey Havens, ruling by force and fear, although still able to put on a mask of wisdom and fairness. In those elf-lands he was called the Dark Lord, the Enemy, and many fled through the Havens on to Valinor. But Gil-galad and Elrond remained, keeping up the places Sauron could not yet overcome. The men of Numenor then rose up, and Sauron went on to become a hostage in their land, his wiles overcoming his captors, making them fear death and causing terrible consequences even he could not foresee, in a short-lived war of part of the Numenoreans against the Valar. He had not expected his own death in all this, and his spirit fled back to Middle-earth.
There he eventually built a new form, no longer able to make one that appeared fair. Once again he took up the Great Ring. Few of Elves or Men could endure his Eye's malice. Once again he built up power, warring on the Elves and the survivors of the Numenoreans faithful to the Valar, the men of Westernesse. Many other Numenoreans he took to himself as servants, the Haradrim. After Minas Ithil fell to Sauron and Osgiliath held and threw back his forces, the Last Alliance was created by Gil-Galad and Elendil against Sauron. After a terrible war, ending with Sauron in a hand-to-hand fight against Gil-Galad and Elendil leading to the deaths of all three, Isildur, Elendil's son, cut off the One Ring from Sauron's hand. He refused to hand it over to Elrond and Cirdan (Ring-bearers both) standing nearby who wished it destroyed. Instead he kept it as were-guild (a payment by a murderer to the family of the victim) for his father Elendil and brother Anarion, and as an "heirloom" of his house. For the first time, Sauron no longer had control of the One Ring. Much of his will and strength continued to survive in this metal band, so that he was not himself destroyed. His spirit hid in waste places for long years before he again took shape.
On his way back to Eriador in the Misty Mountains, orcs attacked Isildur's party in his unguarded camp, for he believed all his foes overthrown. Most of his people, and three of his sons were killed in the attack. Fortunately for his line, his wife and youngest son, Valandil, were in Imladris where he had left them when he went to war. Three of his people escaped, including Ohtar, his esquire who carried Elendil's sword which had broken, which eventually went to Isildur's heir, Valandil. Isildur himself escaped the attack by putting on the One Ring and becoming invisible. The orcs, however, tracked him by scent. He tried to escape by swimming the river, where they could not smell him, but the Ring betrayed him and avenged its maker by slipping from his finger. The orcs killed Isildur.
The One Ring now had no finger and no mind to influence to help it return to Sauron, sitting in the Anduin river. And Sauron had no finger to place it upon.
The Ringwraiths with the Nine Rings returned to Minas Ithil, and a plague came on the winds from the east killing the King of Gondor, his children, and many of the people. Minas Ithil was renamed Minas Morgul, warring with Minas Anor, renamed Minas Tirith. Deserted Osgiliath became a place of ghosts. The Stewards took up the defense against the forces of the East, aided openly by the Rohirrim and in secret by ancient powers to the north which could not be beaten until Sauron himself should come again.
Meanwhile, the Three Elven-rings worked. Elrond kept the Ring of Sapphire, Ring of Air, Vilya, strengthening Imladris/Rivendell where the line of the High Kings was fostered and trained, and elf-life continued as before. Gil-Galad may have worn it first, then given it to Elrond. The Noldor Lady Galadriel, fairest and mightiest of the Elves still in Middle-earth, kept the Ring of Adamant, Ring of Water, Nenya , in Lothlorien, where she was married to Celeborn of Doriath of the Woodland Elves, also protecting the land. The Red Ring of Fire, Narya, Narya the Great was held at this time by Cirdan the Shipwright of the Grey Havens.
Sauron rose again in what had been Greenwood the Great under the Elven-king Thranduil, known to Bilbo. Under Sauron, the forest became known as Mirkwood, from the shadow of Sauron. Thranduil and his people held off the shadow where they could, not knowing the true nature of the Sorceror of Dol Guldur.
As his shadow rose, the Istari appeared in the west of Middle-earth. Cirdan knew they had come from Valinor, and told only Elrond and Galadriel of this, although later it was told among the Elves that they were messengers from the Valar to contest Sauron by moving the Children of Iluvatar and all living things of good will to do great deeds. Cirdan passed on the Ring of Fire to Gandalf, so secretly that only Elrond, Galadriel, and Cirdan knew of it. The Red Ring of Fire rekindled hearts. Gandalf refused to have any ties or allegience except to the Valar who had sent him, so he refused to lead the White Council./Council of the Wise, the leadership of which went instead to Saruman who soon began to study the Rings.
Gandalf looked in on the shadow in Mirkwood, and discovered it was not a Ringwraith, but Sauron himself, and that he was gathering the Rings, hunting the One, and looking for the heir(s) of Isildur who might have the One Ring. The One was believed lost in the Anduin and then on out to the Sea.
But actually, hobbits took control of the One Ring, the hobbits being a race overlooked but of such strong constitution that they could not easily be turned into wraiths. The first to find the One Ring was a hobbit of the Stoor house, named Deagol, while out fishing. He saw the gleam in the mud under the water and decided to keep it. But his brother Smeagol saw the ring, and required it of him for his birthday present, but Deagol refused. Smeagol killed his brother for the bright and beautiful gold. When he returned to his family, the One Ring gave him power according to his stature, so that he could sneak invisibly and learn secrets of others for blackmail, until he was exiled from his people. They nicknamed him Gollum, for a self-pitying noise he made in his throat. He had always been interested in the roots of things and went to check out what those of the Misty Mountains were like. But he found no roots and was bottled in the caverns by orcs. His life was stretched by the Ring, as it waited its chance to return to Sauron.
The Ring could not go to a greedy orc and hope to go towards its Master, but when new creatures entered the caves, mostly dwarves, it dropped from Smeagol's finger in order to find a new carrier. But the power of the Valar caused the scheme to go awry, so that another hobbit of a good nature, named Bilbo Baggins, picked it up instead. He carried it eventually to one of the most protected regions in Middle-earth, called the Shire. It was far from Mordor, protected by Rangers who were men of Westernesse, and a study place of Gandalf's. The Istari, Gandalf, eventually learned what his friend, Bilbo, really had, and determined that events should cause the Ring to be unmade in the place where Sauron had originally made it.
By that time, Bilbo's life had been stretched unnaturally long so that he was not strong enough for such an arduous adventure. Instead, his orphaned nephew and adopted heir, Frodo Baggins, greatest of the hobbits in his nature, became the hope of all Middle-earth. He alone was capable of bearing it for a very great length of time without being taken over by it, a feat not even the Istari could do. The other beings of power, save one, could not keep from using it. That one, Tom Bombadil, did not have power outside his own lands, although within them the ring had no power over him, and if he had kept the Ring, he would have been careless with it, and if he still had it, Sauron would have eventually bent and twisted him as he did the trees and land. Only a hobbit could do the deed required.
Frodo had the indispensable help of SamWise Gamgee , who had the love, loyalty, and responsibility needed to bear the Ring during Frodo's capture, return it in an uprecedented action, and to help Frodo survive the journey. Also, Frodo unwittingly caused the Ring to destroy itself when the time came.
For it was the Ring that destroyed itself in the end, due to the oath of one hobbit to another. To Frodo, Gollum swore by the One Ring, which he called "The Precious", that he would never let Sauron have the Ring, and he would serve the Master of the Ring. Frodo warned him that the Ring was treacherous and would twist his words. Frodo stated to Gollum that the Ring could make him leap from a cliff or jump into fire if he broke his oath by harming the Master of the Ring, the one on whose finger it was. At the Crack of Doom, Frodo claimed the Ring as its Master, putting it on his finger. Gollum bit off Frodo's finger with the Ring still on it, harming Frodo. As Gollum danced in joy, the Ring did its pre-set duty against the one who harmed the one whose finger it was on, forcing both the actions Frodo had spoken of, leaping from the cliff and into fire. The Ring did not seem to understand that the finger it was on was not attached to Frodo, and would not have expected to be destroyed. Gollum, with Frodo's bitten-off finger in hand, fell into the Crack of Doom, destroying the Ring, Sauron, and the works of Sauron in which the Ring and its controlled Rings had been involved.