The basic word is "crying." They describe different ways of crying. "Weeping" emphasizes tears. Tears are drops, almost like water, from the corners of the eyes. "Weeping" is not a sound; a person can weep silently. "Sobbing" emphasizes the sound made by the chest. The chest heaves. There's a repeated, low-pitched grunting or groaning. There is a gasping sound. Oddly, it sounds a little bit like a belly-laugh. "Wailing" emphasizes high-pitched sounds from the throat, almost like screaming.

they all mean crying. Wailing is crying/ mourning loudly - perhaps even making more noise than tears. Sobbing is crying profusely. Weeping is a close synonym for crying, one might weep silently. I would use wailing to convey a lot of noise. Sobbing and weeping are very close, buy sobbing is more profuse than weeping.