The references in this movie are deeply religious, but the message is accessible to a much wide audience.
I wanted to explore the more esoteric meanings from a religious Jewish point of view.
Honestly this movie bothered me a lot and I've cried a bit thinking about it.
The opening scene is the snowy shtetl of late 1800s Czechoslovakia. The somewhat ignorant but still religiously faithful husband and wife have encountered a dybbuk, meaning a demon in human form. (The dybbuk is the one that looks like a rabbi.)
The husband doesn't realize that the religious-seeming man is actually a demon, but the wife knows it right away, and she drives an icepick into his heart to smite him.
Her husband is afraid that they'll get in trouble because she's killed a man, but she knows better, and she slams the door and spits: "Good riddance to evil."
Fast forward to America in ~1970. The Coen brothers present us with the same situation; a somewhat ignorant but religiously respectful Jewish husband and wife, and a dybbuk - so to speak. Not a real demon, just a bad person who lies, cheats, and steals pretending to be good.
Without giving away too much, this "friend of the family" is trusted by everyone, and the rabbi calls him a "tzaddik," but he is stealing the wife of a trusting man, and actually tries to hug him and pretend it's no big deal.
Larry Gopnik is a physics professor and he understands the world of rational thinking. (The dybbuk in the Old World also talked about rationality.) He can't comprehend what God wants, what his wife wants, his kids.
Larry has lost complete control of his family. His son is getting high. In this, one of the most visually captivating scenes from the movie, his son stands in front of an approaching tornado, and the yeshiva teacher struggles to open the tornado shelter.
Larry goes to rabbi after rabbi for help. He begs to know what God wants from him: "I haven't done anything." The most well-known rabbi tells him that his religious feelings are like a "toothache" -- eventually they will go away and he'll be happy again.