Before the Island
At their wedding, Sun gave this wedding ring to Jin as a token of her commitment to him. ("...In Translation")
On the Island
Jin gives his ring to Locke. ("This Place Is Death")
After crashing on the Island, Jin wore this ring for over 100 days while living with the survivors of Flight 815. After narrowly surviving the explosion of the Kahana, Jin joined Sawyer's group. From them he learned that his wife had already left the Island, and that John Locke, a member of that group at the time, was going on a mission to get her, and the other five who left, to return to the Island.
When a dying Charlotte warned Jin to never let his wife come to the place of "death", Jin began to doubt whether it was good for his wife to return. Before Locke descended in the Orchid well to turn the frozen wheel, Jin gave his wedding ring to Locke, insisting that he take the ring to Sun. This would prove to her that Jin had been killed in the Kahana explosion and that his dead body had washed ashore, so she would have no reason to return to the Island. Taking the ring, Locke agreed, before descending into the chamber, turning the wheel, and leaving the Island. ("This Place Is Death")
Off the Island
Ben gives Sun the ring. ("This Place Is Death")
Locke kept the ring in his pocket throughout his time on the real world. In honoring his promise, he did not visit Sun at all, instead allowing her to continue believing Jin was dead. However, John was soon strangled to death by Benjamin Linus, who took the ring from Locke. ("The Life and Death of Jeremy Bentham")
Ben then used the ring to his own advantage, showing Sun the ring as proof that Jin was alive. After finding out this fact, Sun was eager to return to the Island, desperate to see her husband again, and eventually boarded Ajira Airways Flight 316 and crashing back on the Island. ("316")
On the Island again
Sun returns the ring to the rightfull owner. ("The Candidate")
The ring as seen at Lost: The Auction. "We Will Never Be Apart".
When Sun and Jin were finally reunited they were thrown in the Hydra cages by Widmore's team. There they had time to kiss and talk about their daughter. As a token of her everlasting love, Sun returned Jin's ring to him. Later that same day both died, in an exploded submarine.
Till death do us part. ("The Candidate")