Dear Clement:

Most Benedictines are priests, some are non-ordained brothers, but all of them are monks (brothers). Thus, there are priestly monks (brothers) and lay monks (brothers).

The religious state is in itself neither lay or cleric. It is a separate state in which some priests and some laity make vows to the Evangelical Counsels of Poverty, Chastity, and Obedience in a religious order.

While some men who are already priests may decide to transfer from being a Diocesan Priest to a Religious Priest in an Order, most often men join the Order as laymen. Then if the Order permits it, the lay Monk is allowed to receive seminary training in preparation for ordination as a Priest. A Religious Priest lives in the monastery with everyone else. Lay Monks may also have seminary training or other advanced educational training, but are not ordained.

Contrary to popular opinion, it is not necessary to be a priest to be a monk nor is being a Lay Brother necessarily "on the way" to be a priestly monk. Many monks never become priests. St. Francis of Assisi, for example, was not a priest.

Some people seem to think that going on to be a priest is a more complete expression of Religious Life, but this is not so. Remaining a lay monk has the same dignity in religious life as a priestly monk -- one is not "higher" than the other in Religious Life.

Unless an order is a "Clerical Order" which means it is mostly priests (like the Jesuits and Dominicans) because the purpose of the Order includes administering the Sacraments on the mission field or in local parishes, preaching in Mass, etc., a Religious Order usually ordains only enough men as are needed to take care of the Sacramental needs of the Order. The rest remain lay monks.

Generally everyone in a Religious Order, at least non-clerical orders, are called "Brothers" even if they are priests. The habits worn by Religious Priests and Lay Brothers normally look the same. Thus, one cannot tell if a monk is priest or lay from looking at their attire. Priests in other orders, like the Jesuits, wear the usual clerical collar and are always referred to as "Father." 

The habit of the Benedictines, for example, are the same whether the monk is a priest or lay, though I think the Benedictines refer to their priests as "Father."

The Order founded by St. Jean-Baptiste de La Salle is called the Institute of Brothers of Christian Schools, or just Christian Brothers. I believe the order is still all lay brothers (no priests). It was the first religious community of men in the Church not to include clergy.  The story goes that St. Jean-Baptiste de La Salle asked one brother, Henri L'Heureux, to study for the priesthood. It was the St. John-Baptiste's intention to have Bro. Henri take over the supervision of the Institute.  Unfortunately, Bro. Henri died unexpectedly before being ordained. St. Jean-Baptiste took this as a sign from God that the order should remain as a society of lay brothers.

The Christian Brothers, and monks of many other orders, may look like priests because of their habits (as mentioned there is generally no distinction in habits between priest brothers and lay brothers), but as far as I know, the Christian Brothers are still a non-clerical order of lay religious.

A little bit of history: St. Jean-Baptiste received a great deal of hostility from bishops and established orders. Later in life St. Jean-Baptiste had his priestly faculties suspended on some trumped up charge.

Such difficulties are not unusual. Bishops, other established Religious Orders, and the laity are often hostile and downright bigoted against new Orders. I think God uses this human weakness as a test to the founders of new Orders. If the founder cannot persevere through the hostility and even persecution sometimes coming from bishops, priests, established religious orders, and laity then he has no business founding a new Order. St. Jean-Baptiste did persevere.

God Bless,
Bro. Ignatius