Wikipedia
A gremiale, sometimes anglicized as gremial, is a square or oblong cloth which a bishop, according to the " Cæremoniale Episcoporum" and " Pontificale", should wear over his lap, when seated on the throne during the singing of the Kyrie, Gloria and Credo by the choir, during the distribution of blessed candles, palms or ashes, during the washing of feet in the Mass of the Lord's Supper, and also during the anointments in connection with Holy orders.
The gremiale is never used during pontifical Vespers. The primary object of the gremiale is to prevent the soiling of the other pontifical vestments, especially the chasuble.
The gremiale used during the pontifical Mass is made of silk. It should be decorated by a cross in the centre, and trimmed with silk embroidery. Its colour must correspond with the colour of the chasuble.
The gremiales used at other functions are made of linen, to facilitate their cleansing in case they be soiled.
Little is known of its history; apparently its origin dates back to the later Middle Ages. The Roman Ordo of Gaetano Stefaneschi (c. 1311) mention it first (n. 48); soon after it is mentioned in the statutes of John Grandisson of Exeter as early as 1339, In earlier times it was used not only any bishop but also by priests. It is not blessed and has no symbolical meaning.