If you're so upset you're crying loudly, taking in big gasps of air and heaving your chest, you are sobbing.
Picture a child on the playground who has fallen and can't find his or her mother. The child's face is red and streaked with tears, and he or she is gasping from crying so hard. This poor little boy or girl is sobbing uncontrollably. Sometimes, while trying not to cry, you might let out a single sob. A "sob story" is a story meant to make you feel bad for someone, so bad you might sob for them.