"The lesser rings were only essays of the craft before it was full-grown, and to the Elven-smiths they were but trifles..."
- Gandalf in The Fellowship of the Ring, "Shadow of the Past"
Magic rings are only mentioned a handful of times in The Lord of the Rings; magic rings, or lesser rings, are rings with magic powers, but of less potency. They were made around the same time as the Rings of Power and were, presumably, many of them since "In Eregion long ago many Elven-rings were made". Gandalf states that though the Elves who forged them thought little of them, he still thinks they are dangerous for mortals.
They are known as the lesser rings of the Rings of Power, or simply magic rings when referring to the rings forged by the Elves as "essays in the craft"- they are said to bear no gem (similar to the One Ring). It is unknown if they had marks or inscriptions on them, or having the ability to produce fiery letters like the One Ring.
In the ancient days of the Second Age, before the Fall of Númenor, Sauron came to the elven smiths of Eregion declaring he was sent by the Valar to bring gifts and knowledge to them. Using a fair guise, he deceived the elves in helping him forge magic rings, especially Celebrimbor. With the help of the greatest Elvish craftsman, Sauron was able to continue with his plans for Middle-earth by making of magic rings. before the Rings of Power were forged, only lesser magic rings were made by the craft taught by Sauron as magic ring-forging was just in its infancy. Though the lesser rings were quite useful, they did not have the strength, abilities, or influence of power as the yet-to-be-forged Rings of Power. Despite the current limitations, the Elves used these rings to aided them in the use of spells and magical craft and were thus widely popular. Sauron, then known as "Lord of the Gifts" claimed he wished to make Middle-earth as fair as Valinor, and it seemed he convinced the elves to think these rings could do so. And as time went by, Sauron's and Celebrimbor's experimentations with the rings led to the forging of rings with far greater power and abilities. When magic ring-forging reached its highest, Sauron felt confident that he could make his magnum-opus, the One Ring.
After the forging of the One Ring, Sauron began drawing the lesser rings to himself. This was not difficult, for the One Ring gave him the strength to subdue the wielders of these powerful artifacts. Once gathered, Sauron could disperse them among his captains, making them all mighty champions whom he could back up at will with his own strength via the magic ring connection. After the Last Alliance of Elves and Men and the First Fall of Sauron, these rings were dispersed through the land. Some were taken from dead warriors, others, borne away by the frightened, still living servants of Sauron, others were lost and little is known of what happened to them until the War of the Ring.
At the time of the War of the Ring, Sauron gathered again the lesser rings to himself. But this time, he was unable to retrieve all of them as many of them were lost and his attention was more focused on claiming the Rings of Power and his One Ring. The few he did find were used to advance his own abilities. Though not stated in the books or movies, it seems reasonable to believe that some were given again to his commanders as well.
Their ultimate fates are unknown, nor is it known whether or not their powers were bound to the power of the One Ring. If they were, then their power would have failed with the destruction of the One.
The full extent of the rings' power is unknown, but Gandalf says that "none are to be used lightly". They were, to an extent, like the One Ring when it came to temptation, but of a lesser form. They would give a wearer strength, and the power to dominate other wills. Some could also make their wearers invisible.
It is also believed (though again, not stated) that these rings also had certain fields in which they were strongest. For example, one ring may rule over stone, while another controls ice. This, however, is not known fact, and only stems from what is known about the Rings of Power, assuming that the lesser rings are much like them.
Also in the 1995-1997 Middle-earth Collectible Card Game several cards describing and depicting lesser rings were released, for example "The Oracle's Ring", "Wizard's Ring" and "Magic Rings" of different power (lore, stealth, courage, etc.).