It's actually only mentioned very briefly, after Romeo's been banished. Juliet sends the Nurse to find Romeo, and sends the ring as a token of the fact that she does not think of Romeo, even though he has killed Tybalt, as an "old murderer", but as her "true knight":
O, find him! give this ring to my true knight,
And bid him come to take his last farewell.
In the next scene, the Nurse and Friar Laurence try - without much success - to counsel Romeo and calm him down, but he actually is wavering about whether he can face Juliet again until he receives the token from the Nurse, right at the end of the scene: these are the Nurse's line:
Here, sir, a ring she bid me give you, sir:
Hie you, make haste, for it grows very late.
And Romeo is brought back to life by the thought that Juliet still loves him.
How well my comfort is revived by this!
Secondly, and importantly, the ring represented the female genitalia to the Elizabethans (so that's what that last act of "The Merchant of Venice" is about). There's also some sexual play going on: particularly as, when Romeo does come to Juliet's room that night, they consummate their marriage.
Hope it helps!