Doing something peacefully means acting in a calm, easygoing way. When a cranky baby is finally sleeping peacefully, it's a huge relief for everyone.
You can also use this adverb to describe things that happen with a total lack of violence: "The countries settled the dispute peacefully instead of going to war." Anything you do peacefully has a sense of gentleness and serenity about it, whether you breathe peacefully while you do yoga, or smile peacefully at your bickering brothers. The root of this word means "to fasten," in the sense of a peace treaty binding people together.