If you do something languidly, you do it slowly, without haste, as if you're a Southern belle without a care in the world lazily sitting around on the veranda on a hot day.
The adverb languidly comes from the Latin word languere, meaning "be weak or faint." When you do something such as run or climb stairs languidly, you probably do appear weak or sluggish. The word has a peaceful and relaxed connotation, however — if you do something languidly you aren't trying very hard and you aren't worrying much about it. You may chat languidly with your friends on a summer day while sipping tea on the patio.