The Glass Menagerie is fragile and delicate, just as Laura. This fragility is manifested physically in the glass; as Laura says, "If you breathe, it breaks!" Yowzah. It’s also really beautiful, as those of you with unhealthy glass animal fetishes may have already known. Laura has the same kind of beauty – the translucent, other-worldly, delicate kind.
Then there’s this unicorn business. When Laura talks about the unicorn, she reveals that it is her favorite glass animal, that it is unique from all the other horses because of its horn. Not that Laura has a horn, as far as we know, but she is really different from most girls, as Jim recognizes. Part of what separates her from the pack is her off-the-charts shyness. So when Jim makes her dance with him, and the horn breaks off the unicorn, Laura calls it a "blessing in disguise" – she is being made to be a normal person. However, when Jim turns out to be engaged, she gives him the unicorn, in a tragic sort of "look what you broke in me" kind of way.