Elie Wiesel had one main purpose for writing of his experiences during the Holocaust (in his novel Night). According to his introduction, Elie knew that the Holocaust and the period of time which surrounded it "would be judged one day." Given that he recognized this, Elie believed that someone would need to "bear witness." For him, the someone was him.
Elie did have a fear though: that the testimony he would provide would not be received well. He knew that the only people who would truly understand were those who had lived through and experienced life in the concentration camps. Even though Elie realized that they, readers, would not "know" the reality behind the Holocaust, he hoped that they would "understand."
This understanding Elie desired was problematic given his language. He worried that he would not be able to help those who did not experience it because the words he chose may be wrong. Regardless of this barrier, Elie stood strong. He refused to stay silent (like the world when it happened--given the first title of his novella: And the World Remained Silent).
Therefore, his purpose in writing Night was to not stay silent and bear witness to the Holocaust.
A writer's purpose can usually be found in the themes that he develops in his work. In Elie Wiesel's Night, there are several themes that run throughout the work.
When Wiesel writes about the cruelty of the Nazi soldiers in the concentration camps, he is developing the theme of man's inhumanity to man. This becomes poignantly evident when we see changes in Elie himself. At one point he tells the reader that because of his hunger and deprivation, he had become nothing more than "a stomach."
However, Wiesel also develops a counter-theme of kindness under severe conditions. Despite the suffering and ever-present threat of death, there are still moments when people are kind and giving. At one point, a young violinist named Juliek, who somehow managed to hang on to his violin in the midst of forced marches, plays beautifully for the exhausted and dying prisoners, only to die himself soon after. Elie's father continues to give Elie his own rations as he faces starvation himself. Sometimes prisoners risk their own safety to give comfort to those who are suffering.
Wiesel's main purpose is no doubt to ensure that the world knows what happened in the camps. His own characters persistently denied their fate early in the book, refusing to believe the things they had heard about the Nazis' plans for them, refusing to believe that such terrible things could happen to them in the twentieth century. Wiesel wants to make sure that the rest of the world doesn't make that mistake.