What does the ring symbolize in A Doll's House?

I assume that you're referring to the Ibsen play A Doll's House rather than the Katherine Mansfield short story of the same name, as a ring is an important symbol in the play rather than the story. For Nora, her wedding ring is a symbol of a failing marriage and all that that entails. Like most women of the time, Nora has little freedom as a married woman. The wedding ring that she wears increasingly becomes a symbol of her subservience; that she's regarded by a male-dominated society as being little more than the property of her husband, Torvald. The ring becomes almost like a chain, manacling her to a man whose true selfishness and narcissism are gradually revealed.
But more than anything else, Nora's wedding ring is a perfect symbol of her marriage because it is superficially pretty, something to be shown off to the whole world, and yet which has no real substance to it. Her marriage, like the ring, is ultimately just a bright, shiny bauble of purely economic value.