The faerie rings are special, faerie-work rings that allows the wearers to communicate through each other's minds. The rings came in a pair and are golden, with a leaf pattern.
The rings were, at some point, taken from the faeries and placed on display at the New York Institute's library. The rings were kept there until 2007, when the Seelie Queen, who claimed that they had belonged to her father, tasked Clary Fray to retrieve the rings and return them to her, on the condition that she would help Clary find the missing Jace Herondale should Clary fulfill her request.
Clary indeed stole the rings from the Institute with the intention of giving them to the Queen, until she saw Jace and Sebastian breaking into the library. She then made the decision not to give the rings to the Seelie Queen, realizing that she was not going to get anything useful from her anyway. Clary then gave one of the rings to Simon Lewis while she kept the other, as a means of communication while she went off with Jace to find out what he and Sebastian were planning together.
Unbeknownst to them, the Queen had expected that Clary would keep the rings for herself to communicate with her friends and was able to secretly listen in on her conversations with Simon. She intended for Clary to take it where Sebastian was, hoping he would recognize the ring and get ahold of it so they could communicate. Jonathan stole it from her one evening, while she was high on faerie drugs at a bar in Prague, and immediately contacted the Seelie Queen. Eventually, Clary found the ring in Sebastian's room and, to stop Jonathan from being able to further communicate with the Queen, destroyed her ring.
Because she had destroyed the one she wore, they believe that the one Simon has no longer wields powers. Nonetheless, Simon continued to wear the remaining ring as a symbol of his connection to Clary. Just before Asmodeus took Simon's memories and immortality, as well as his body from Edom to the mundane world, Simon tossed the ring to Clary, who then began wearing it. After Magnus helped restore some of his memories of the Shadow World, however, Simon remembered enough of his and Clary's friendship to know that the ring meant something to them.