It depends on which part of the story that you are referring to. Della is crying at the beginning of the story AND at the end of the story. In the beginning of the story, Della is crying out of legitimate sadness. She and her husband, Jim, are poor. Really poor. They have exactly $1.87 to their names. Della isn't unhappy that she and Jim are poor. Their love for each other is enough to keep her and Jim happy despite their financial situation. Della is sad that she is unable to buy Jim a decent Christmas present. Christmas day is the next day, and she desperately wants to buy Jim a present, but just can't afford it.
In order to afford a present for Jim, Della decides to sell her hair. Jim's watch and her hair are the only two things of value that they own. With her hair cut, Della can now buy Jim a chain for his watch.
That evening Jim and Della exchange their presents, and that is when Della starts crying again. Jim's gift to Della is a pair of beautiful combs for her hair that she has been wanting for a long time. Now that her hair is cut so short, she cannot use them. Della cries at this point for a couple of reasons. She cries because she's happy. Jim, out of tremendous love, sold the watch to afford the combs. She's happy that she has the combs, and that Jim loves her so much. She also cries out of sadness. Della finally has the beautiful combs, but she no longer has enough hair to use them. Perhaps Della is crying because she thinks that she has disappointed Jim in some way. He gave her the combs, but she cannot immediately use them.
Della cries for a variety of reasons throughout the story, but there is no doubt that Jim and Della's love for each other is the main point of the story.