These were made by the Elven smiths of Eregion. The Seven and the Nine were made with Sauron's assistance, the Three were made by Celebrimbor himself, and the One was made by Sauron alone, in secret.
It was in Eregion that the counsels of Sauron were most gladly received, for in that land the Noldor desired ever to increase the skill and subtlety of their works. Moreover they were not at peace in their hearts, since they had refused to return into the West, and they desired both to stay in Middle-earth, which indeed they loved, and yet to enjoy the bliss of those that had departed. Therefore they hearkened to Sauron, and they learned of him many things, for his knowledge was great. In those days the smiths of Ost-in-Edhil surpassed all that they had contrived before; and they took thought, and they made Rings of Power. But Sauron guided their labours, and he was aware of all that they did; for his desire was to set a bond upon the Elves and to bring them under his vigilance.
Now the Elves made many rings; but secretly Sauron made One Ring to rule all the others, and their power was bound up with it, to be subject wholly to it and to last only so long as it too should last. And much of the strength and will of Sauron passed into that One Ring; for the power of the Elven-rings was very great, and that which should govern them must be a thing of surpassing potency; and Sauron forged it in the Mountain of Fire in the Land of Shadow. And while he wore the One Ring he could perceive all the things that were done by means of the lesser rings, and he could see and govern the very thoughts of those that wore them.
But the Elves were not so lightly to be caught. As soon as Sauron set the One Ring upon his finger they were aware of him; and they knew him, and perceived that he would be master of them, and of an that they wrought. Then in anger and fear they took off their rings. But he, finding that he was betrayed and that the Elves were not deceived, was filled with wrath; and he came against them with open war, demanding that all the rings should be delivered to him, since the Elven-smiths could not have attained to their making without his lore and counsel. But the Elves fled from him; and three of their rings they saved, and bore them away, and hid them.
Now these were the Three that had last been made, and they possessed the greatest powers. Narya, Nenya, and Vilya, they were named, the Rings of Fire, and of Water, and of Air, set with ruby and adamant and sapphire; and of all the Elven-rings Sauron most desired to possess them, for those who had them in their keeping could ward off the decays of time and postpone the weariness of the world. But Sauron could not discover them, for they were given into the hands of the Wise, who concealed them and never again used them openly while Sauron kept the Ruling Ring. Therefore the Three remained unsullied, for they were forged by Celebrimbor alone, and the hand of Sauron had never touched them; yet they also were subject to the One.
The Silmarillion, "Of the Rings Of Power and the Third Age (in which these tales come to their end)"