From Wikipedia, on the traditional usage of the rite:
Traditional Form:Where the 1962 Missal is used, the Asperges is done before the principal Mass on Sunday. The Asperges is so called from the words intoned at the beginning of the ceremony, taken from Psalm 51, throughout the year except at Eastertide, when Vidi aquam, with Psalm 117, is intoned. It precedes every other ceremony that may take place before the Mass, such as the blessing of palms or of candles. It is performed by the celebrant priest wearing vestments of the liturgical color of the day. It is omitted when the Blessed Sacrament is exposed, though many rubricists think that the sprinkling of the altar only, not of the congregation, should then be omitted. After intoning the antiphon the priest recites the psalm Miserere or Confitemini, according to the season, sprinkling first the front and platform of the altar, then himself, next the ministers and choir, and lastly the congregation, usually walking through the main part of the church, though he need not go beyond the gate of the sanctuary or choir. The ceremony has been in use at least from the tenth century, growing out of the custom of early antiquity of blessing water for the faithful on Sundays. Its object is to prepare the congregation for the celebration of the Mass by moving them to sentiments of penance and reverence suggested by the words of the 51st psalm, or by impressing on them that they are about to assist at the sacrifice of our redemption as suggested in the psalm used at Easter time.Both the Asperges and the Vidi aquam are structured like the Introit of the Tridentine Mass: 1st verse (which includes Alleluias in Eastertide), 2nd verse (always from the Psalms), Gloria Patri (omitted in Passiontide), and then the 1st verse again.