The significance of the title of the story "The Student" by Anton Chekhov is two-fold.
First, the character of the student is that of 22 year-old Ivan Velikopolsky; a student of the clerical academy and son of a sacristan. Ivan is returning home from shooting with mixed feelings and thoughts in his mind. As a young man, he is discouraged about the current situation in his life, as well as his country; it is cold, society is broken, poverty is everywhere and, like Chekhov poignantly specifies
And he did not want to go home
The event that moves the plot forward and starts the epiphany that would render significance to the title of the story is that Ivan stumbles upon a campfire kept by the widow Vasilisa and, for the sake of conversation (and as a student of religion) he related the story of how Peter denied Jesus three times as Jesus had predicted; the connection came from the fact that Peter was standing at a shared campfire when this occurred.
The instant reaction that the story caused in the widow Vasilisa led the student Ivan to make a massive realization:
The past," he thought, "is linked with the present by an unbroken chain of events flowing one out of another." And it seemed to him that he had just seen both ends of that chain; that when he touched one end the other quivered.
Hence, the significance is that even he is a student he served as a teacher to the widow Vasilisa and made a difference in her life. The second significance is that the student has learned yet another lesson; not only what he learned at seminary but also the other metacognitive realization of the continuity of life, history, and the experience of being human and spiritual at the same time.