A waistcoat (prounced more like "weskit") was a sleeveless vest as you see in the movies etc. Entirely taking off your tailcoat (worn over the waistcoat) was somewhat casual, probably a little too informal for most occasions on which Marianne would be likely to meet the Col.; but leaving your tailcoat unbuttoned, and so exposing the front of your waistcoat beneath, was quite common...