Screwtape is a senior tempter in a department of Hell and writes letters to a junior tempter, Wormwood, instructing him how to tempt his subjects -- a woman and her son.
Basically it's a very clever expose of inter-personal relationships and the way we can live together easily, or with friction.
In a way it's a dramatisation of the way our subconscious works, rationalising, persuading us to do the things we want, and he does it by personalising our subconscious as though there was actually a little Devil sitting on our shoulder saying, "Go on, you know you want to."
If you want to understand how your mind works, you could do a lot worse than read Screwtape.
Dodgy gives a good answer but there is more to it. It's about the spiritual battle that is around us that we don't see. The letters are from an senior demon who is the uncle of a younger demon. He teaches him how to use criticalness and even good things to his advantage so that people will turn from God or be deceived in self righteousness. It is very insightful and easily read a chapter at a time.