Bye-bye is a warm, informal way to say goodbye. Adults say it to small children, and children say it back; friends and family may say it too when they are being cute or close. You would not put it at the end of a serious letter or a job email. People often wave when they say it. You can also use it as a light noun: “Say bye-bye to Grandma.” In very casual talk it can mean you are done with something — “That’s bye-bye to my weekend plans” — but the main job is still the goodbye. Though actually adults say it to each other too when they are being playful, not only to children.