Celebrations in the absence of a priest.

The ideal case throughout the world is for each group, each parish to have each Sunday at least one Mass that the faithful can attend. Unfortunately this is not always possible. In some areas of the world there are simply not enough priests available to serve all of the faithful. At other times a parish which has a faithful priest is suddenly without one due to illness or other unforeseen circumstance. Many times this can be dealt with by contacting a substitute to come in and say Mass. Sometimes this is not possible.
The preferred action in this case is for the faithful to visit surrounding parishes to attend the Masses there. The Congregation for the Doctrine of the Faith, and the bishop conferences realize this is not alway possible. Surrounding parishes may be too far away. Masses there may have already taken place, or it may not be possible to inform the faithful in a timely manner, so that they could arrange to attend elsewhere. In that case Sunday Celebrations in the Absence of a Priest can be authorized.
It is up to each bishop to decide when it is appropriate for this liturgical practice to be done. Most diocesan bishops issue a directive, based upon the Congregation for Divine Worship's Directory for Sunday Celebrations in the Absence of a Priest. This document lays out the guidelines for this practice.
As might be supposed, in the absence of a priest, the deacon presides as a member of the clergy, befitting his office. He

acts in accord with his ministry in regard to the greetings, the prayers, the gospel reading and homily, the giving of communion, and the dismissal and blessing. He wears the vestments proper to his ministry, that is, the alb with stole, and, as circumstances suggest, the dalmatic. He uses the presidential chair.

If a deacon is not present a lay minister does not preside, but acts as a leader among equals.

The lay leader wears vesture that is suitable for his or her function or the vesture prescribed by the bishop.34 He or she does not use the presidential chair, but another chair prepared outside the sanctuary.35 Since the altar is the table of sacrifice and of the paschal banquet, its only use in this celebration is for the rite of communion, when the consecrated bread is placed on it before communion is given.

The lay leader is typically an Extraordinary Minister of Communion who has received special training to lead a Celebration in the Absence of a Priest. They must be specifically appoint by the pastor, and in some diocese officially designated by the bishop for this ministry.