godliness: a life of genuine devotion to God that shows itself in how a person actually lives.
In the wider Greek world this word named the proper respect a person owed to the gods, to the city and its rulers, to parents, and to the dead. It was mostly about right conduct, doing the customary, honorable things, and a Greek speaker would also have heard in it loyalty to the emperor and the philosophers' ideal of knowing how to serve the divine rightly.
The New Testament takes that loaded word and fills it with new content. It shows up suddenly and thickly in a few late letters (especially the letters to Timothy and Titus, and in 2 Peter), almost as the name for the whole Christian way of life. It is not a feeling tucked away inside a person, and it is not mere outward respectability either; it joins true belief about God to the conduct that matches it. The letters to Timothy even call it a thing to be trained for, like an athlete training his body, and promise it pays off both now and in the life to come.
What makes the Christian version distinctive is that it is anchored in Christ. One famous line calls it a revealed secret and then quotes a hymn about Christ appearing in human flesh, so the godliness in view is the kind that flows from who Christ is and what he has done. Because of that anchoring, this godliness is not safe, respectable religion: one of the same letters says everyone who wants to live this way in union with Christ will be persecuted.
The word can also point to the practical, lived-out side of that devotion: the concrete godly acts a believer does, or the personal devoutness a person is known for. The same letters warn that this outward shape can be counterfeited, worn as a respectable form by people who have denied the power that is supposed to drive it, or even peddled as a way to get rich.