“Lest” is a considered a formal word and is normally used with “should”. When you do something lest something bad should happen, you do it to prevent the bad thing from happening. The word means “for fear that” or “in order that”.
Here are a few examples. *Hamsa was afraid to say anything lest her boss should scream at her. *Uma refused to run away lest the police should think that she had committed the crime.