Society of Saint Pius X (SSPX):
The Society of Saint Pius X has an irregular canonical status, operating independently of local bishops, and their Masses are considered valid but illicit (describes a situation where a rite is performed correctly in its essential elements, but in a manner that violates the rules or laws of the Church).
The Society of Saint Pius X should not be confused with either of the following:
The FSSP (Priestly Fraternity of Saint Peter), which is in full canonical standing and communion with the Catholic Church, working with local bishops and offering the traditional Latin Mass.
The Institute of Christ the King Sovereign Priest, Latin: Institutum Christi Regis Summi Sacerdotis, abbreviated as ICKSP and ICRSS, is a society of apostolic life of pontifical right in communion with the Holy See of the Catholic Church.
Both are traditionalist Catholic groups that celebrate the Traditional Latin Mass, but the SSPX’s lack of faculties from a bishop puts them in a complicated, irregular situation with Rome.