This ring was first the property of Finrod Felagund that he brought from Valinor.
Barahir rescued Finrod in the DagorBragollach. Finrod gave the ring to Barahir as a token sealing an oath of unabiding friendship and aid.
Afterwards Barahir lead a band of 'outlaws' that fought against the forces of Morgoth in occupied Dorthonion.
Sauron was sent by Morgoth to destroy the band of men. Through trickery and deceit, Sauron learned the location of Barahir's hidden camp. He sent Orcs seek out and killed the last of the 'outlaws'. The Orcs cut off the hand of Barahir, that had the ring on it, as a token to Sauron and Morgoth of Barahir's defeat. Beren, son of Barahir, who had escape the Orcs, hunted them down and killed the Orc Captain and retreived the hand and the ring.
Later Beren went to Nargothrond and requested aid from Finrod in his quest for the Silmaril. Finrod died saving Beren's life fullfiling his oath to Barahir.
The story of the ring was fairly uneventful after that. Beren married Lúthien and passed the ring to their son Dior, Thingol's Heir. Dior passed it on to his dautgher Elwing. Elwing married Eärendil and they passed the ring on to their son Elros, the first king of Númenor.
The ring later became a heirloom of the House of Isildur and was passed down through the age to Aragorn who gave the ring Arwen as token of their engagement!