Discretion = behave depends on situation or don't say anything offensive

Discretion, according to the dictionary, means: "behaving or speaking in such a way as to avoid causing offense or revealing private information" So basically in your sentence it means - be careful (1 don't talk about it to others 2 don't say anything offensive) It might depend on the context. If it is sensitive because private - don't talk about it. if it is sensitive because the issue might be sensitive - related to - gender, race, religion, sex, etc - then means be careful in your choice of words regarding those topics.

This is very formal and means: "Please keep this quiet. Do not tell other people about this."

This means: I am begging you to be discreet about this matter