What is a genre?
This is an interesting question because you ask for a definition, but placed your question under the topic of Elie Wiesel's Night. I will begin with the definition and then go into how Elie Wiesel's book can be an example of one particular genre.
Stated simply, a genre is a category (in this case) of Literature which has a particular form, content, or style. The English word comes from the French word for "kind" or "type."
A category of artistic composition, as in music or literature, marked by adistinctive style, form, or content.
There are many genres just in Literature. Some common ones are the novel, the short story, the essay, the poem, etc. Some genres are general (such as "non-fiction") and some are very specific (such as "lyric poetry").
Night is not a particularly easy work of literature to pin down in genre. However, since I have been asked, I would say that it is a "personal narrative" steeped in nonfiction. We can't quite say it's "autobiographical" because the protagonist isn't the author; however, we can't say it's historical fiction because the story is absolutely based on the author's life (due to his own testimony). There must be a disconnect somewhere.
If you'd like to get more specific, you can look at a few different genres where the book might fit. I suggest it's a "personal narrative" which means it's fairly short, the author fits the primary character of the book, and it contains personal events and information. Other scholars think it's more of an "autobiography" being that it's factually accurate, written by the main character, and tells the story of that life. Some people think it should be described as a "memoir," which is informal and short while it concentrates on emotions more than facts. Perhaps Wiesel's Night is actually a combination of the three. Note the following quote:
It has always been a joy of mine, as an English Teacher, to tell my students that it doesn't matter which genre they pick. If they use evidence from the book, they can ALL be proved.
Never shall I forget that night, the first night in camp, which has turned my life into one long night, seven times cursed and seven times sealed. ... Never shall I forget those moments which murdered my God and my soul and turned my dreams to dust. Never shall I forget these things, even if I am condemned to live as long as God Himself. Never.