The word friar comes from the Latin word frater, which means brother. When the first mendicants went into the streets to minister they were asked who they were, they identified themselves as Brother Francis or some such name.
Many friars are ordained. However, ordination is not a requirement to be a friar or a monk. The reason that friars continue to call themselves such, is that they do not fit into a neat category of religious life. They are not monastic. They are not secular either. They are truly consecrated religious men, but live and work outside of a cloister.
Some Benedictine monks also work outside the cloister, but they are attached to a monastery for life. A friar is not attached to any house for life. This is why he is called a mendicant. He is an evangelical nomad, my term. He goes anywhere in the world where his community sends him.
Friars are part of large religious orders with centralized governments. Monks are part of a monastery and are governed by the abbot of the monastery.
Most friars live in houses called priories, friaries, or convents. Some friars, such as Capuchins, do live in monasteries. But that’s because the Rule of St. Francis allows for some friars to be hermits.
Holy Orders is not part of religious life. It is a sacrament. A friar or a monk is first a Benedictine, Franciscan, Carmelite, Dominican, Augustinian, Trinitarian and so forth. The priesthood is a vocation within a vocation.
In some communities of friars there is no external distinction between the friar who is a priest and the friar who is a lawyer, teacher, cook, or youth minister. For example, you mentioned the Capuchins. The Capuchins call themselves Friars or Brothers. The laity usually makes the distinction by calling the ordained friars, Father. This is not done within the community. In fact, you can have a house of ordained friars with a superior who is a lay friar.
The word lay means someone who is not ordained. A lay person can obviously be in vows. Nuns, sisters and brothers in congregations are lay, but they are religious. They are lay because they are not deacons, priests or bishops.
In my own Franciscan community, we have one religious who is ordained. The rest are lay, but we are not secular. We are in vows. Our superior is a lay brother and the ordained brother is our vicar (second in command). We are considered friars and we use the title Brother for everyone. Our constitutions say that we only ordain as many brothers as we need for the spiritual life of the fraternity. We’re not a clerical institute.
Some monastic communities follow the same system. They only ordain as many men as they need for their community.
It is important to remember that friars and monks can only be ordained when the community approves. There is a vote taken. The results are given to the superior and his council. They vote again. Finally, it is up to the major superior to grant permission for ordination. The major superior has the authority to deny the ordination. In that case, the monk or the friar is still bound by the vows he made and is bound to the community for the rest of his life. The voice of the community is one of the indicators of whether a man has the vocation to the priesthood or not. The superior acts as a bishop of a diocese would. He authorizes the ordination. Without his permission, the ordination is illicit.
The difference is that if your bishop denies you ordination, you can always apply to another diocese. If your religious superior denies you ordiantion, you have no where else to go, because you are bound to the community by perpetual vows.
The power of religious vows should never be underestimated, whether one is a monk or a friar.