When you do something judiciously, you use common sense or good judgment. It's important for juries to act judiciously when they're making a decision about a verdict.
If you're giving someone bad news, it's best to speak judiciously, carefully considering how your words will be received, and when a teacher chooses one student to be his assistant for the day, he should choose judiciously so the others don't get their feelings hurt. The adverb judiciously comes from the Latin iudicium, "judgment," by way of the Middle French judicieux.