Saint Catherine Laboure lived in 19th century France. She was a nun belonging to the Daughters of Charity Order, an order well-known for their starched white coronets.
Looking at the medal, we can find much symbolism in the symbols used. First of all, Mary, our mother, with her hands outstretched to her children shows us the loving and maternal heart that Mary possesses and the eagerness of her arms to gather her children to her breast.
Next, the rays falling from her hands are the graces she showers on the world. In Sister Catherine's vision, she saw many jeweled rings on Mary's hands. Some of them did not shed the glorious light onto mankind. Mary told Saint Catherine Laboure that the gems that shed light were the graces that she gives to those who ask for them and the gems that did not shed light were the graces that are available but are never asked for.
On the reverse side of the medal, the "M" of course stands for Mary. The cross that comes out of the top shows that Mary faithfully stood at the foot of the cross and is the co-Redemptrix.
On this subject, I once heard a priest talking about how the most painful thing that a woman can suffer is the torment of her child. The tactic of breaking a woman with this method was tried and failed on the unnamed mother of the Maccabees who watched as one by one her sons were killed for refusing to give up his faith.
In a very similar way, Mary was tormented by watching the suffering of her Son. The priest ended his sermon by saying that although "Better love hath no man than he lay down his life for his sheep," a similar saying may be: "Better love hath no woman than she give her Son for her children".
Finally, the twelve stars that border the back of the medal symbolizes the twelve Apostles, who, although not at the foot of the Cross, surround and stay close to Jesus and Mary symbolized by the Sacred and Immaculate Hearts.