The ring was first technically seen on Eobard Thawne's hand when he pushed Barry back with his hand. Later, he used the ring in order to open a containment pod with the Reverse-Flash suit within it in order to put the tachyon prototype on it.
While Eobard was incarcerated in the S.T.A.R. Labs particle accelerator, Cisco Ramon questioned him as to how it worked, but he skipped over the question. Thawne was wearing the ring when his ancestor, Eddie Thawne, committed suicide. When Eobard was erased from existence, the ring was as well. 
After Harrison Wells from Earth Two donned the suit to trick Grodd, Barry made a decision to destroy the suit, but he took the ring and asked Cisco to reverse engineer it. Barry wanted to figure out how to put the suit back in the ring, so that he can use it to carry his suit in a more subtle and accessible way.
Unfortunately, Cisco had yet to find a way to compress the suit into the ring, which led him to question a younger version of Eobard Thawne, but received no answer.
KeyThis ring served as a key to at least two things. Eobard was able to feed his suit that was in the ring through a pink slot, and the suit draped over the pod inside. Also, the ring was the only way to access Gideon's operating system.
DC's Legends of Tomorrow
Behind the scenes
- In the comics, three different iterations of The Flash have used a lightning-bolt insignia ring to contain their suit, chemically treated in order to compress it down; these three are Barry Allen, Wally West and Bart Allen.
- It hasn't been explained how he shrinks the suit down in the show as of yet, but it is possible he used Ray Palmer's shrinking nanotechnology as Ray Palmer's brother, Sydney, sold a lot of his tech and it became commonplace around the time of Eobard's generation.