The story indicated that Elisa Allen cried like an old woman. How does an old woman cry and why do they cry like this?
Well, let us be honest; this is a dated statement, calling on sexist ideas about how men and women act. That said, I would say that this refers to the idea of a person—and in this attitude, especially a woman—being unable to contain her emotions. They just overwhelm her. It also refers to crying out of weakness, rather than passion, and crying over loss—perhaps over what might have been, or what is no more—rather than over, say, a current fiery passion. As far as specific expressions or methods of crying, there wouldn't be any, really.
The tears shed by an old woman represents tears that emanate from a life of things that have already past--things that can no longer change, but are set in place forever. It symbolizes regret for those things that should not have been. These tears are wrought from a body that is tired and weak, perhaps lonely and contemplating the end of life and looking back upon those things that cannot be changed. These are impotent tears from a state of exhaustion and in a sense, defeat.