A society of apostolic life is a congregation (of priests, often) that is not under religious vows. This is in contrast to a society of consecrated life which is under religious consecration. Nevertheless, members of societies of apostolic life live communally, under the bond of charity, and in a hierarchal structure seperate from the diocesan (related to it in various ways depending on the specific congregation). Some even could contain communes of married couples or stuff, though it is usually priests. Examples include the FSSP, ICR, and the Legionnaires.
A secular institute is sort of the opposite. They DO take religious vows, but live in the world. They may or may not wear special dress to indicate their Consecrated status and many have regular jobs and go about their daily bussiness, but live discreetly under their institute’s rule. Sometimes they might live with other members of the institute who agree to assosciate in such a way. They have their own moderator and structure of authority, but it is looser than in other congregations and societies and they are usually closely related in various ways to the local diocesan bishop too. Unlike most institutes of consecrated life (with a more seperate power structure and seperate incardination) even diocesan priests can sometimes assosciate themselves with, and take religious vows under, secular institutes.
