The Gwaith-i-Mírdain ('People of the Jewel-smiths'), Elves living in Eregion (Hollin), forged the Rings of Power (except for the OneRing) in the SecondAge (ca. SeA 1500-1590).
Who of them in particular made the lesser ones (which includes the NineRings and the SevenRings) is nowhere explicitly told, but Sauron (in disguise) had taught them how to forge such rings and guided their labours.
The mightiest of those, the three Elven-rings, had been made by Celebrimbor alone and those have never been touched by Sauron.
The OneRing was secretly forged by Sauron alone (ca. SeA 1600), in the Chambers of Fire at MountDoom. But as soon as Sauron put on the OneRing, the Elves realized this and took off their rings so that Sauron could not gain control over them. When Sauron realized that his plan did not work as intended, he made war against the Elves. In this war Celebrimbor was slain, Eregion laid waste the doors of Moria were shut.
During this war Sauron gathered all the remaining Rings of Power into his hands, perverted them and gave them out to the other peoples of Middle-earth: Seven to the Dwarves and nine to Men.