I can’t speak specifically to the Dominicans, but as a soon-to-be lay Franciscan, perhaps I can help a little.
Some religious orders have developed communities whereby people who want to remain ‘in the world’ rather than becoming priests or taking the vows of a nun or monk can nonetheless follow the rule and spirituality of the Benedictines, Dominicans, Franciscans, Carmelites etc. Suitably modified for them, of course.
A period of discernment is involved, usually upwards of a year, before the final commitment is made to join one of these groups. This is probably the ‘postulancy’ referred to.
Members of these “Third Orders” or “Lay Societies” (they go by many names) can pursue secular careers, marry and have families and so on, but usually come together in local groups (Franciscans call these fraternities) for meetings and prayers. And are under the spiritual guidance of the religious of their particular spiritual ‘family’. They might go on retreats together or work together on social projects, be involved in their local parishes and so on.
How this pans out for me as a secular Franciscan, for example, is that the Franciscan rule of (usually fairly extreme) poverty is modified for us - we’re not severely restricted in terms of possessions but strive nonetheless to live simply and humbly.
SFOs have obligations to pray Morning and Evening prayer daily, to attend Fraternity meetings once a month if possible, to contribute materially to whatever projects the Fraternity may have on hand and the like.
