Brothers are members of religious orders who have taken their solemn vows. Examples of religious orders include Benedictines, Dominicans, Franciscans, and Carthusians. To become a brother you ask to come to a certain monastery and be accepted as a novice. Then, as a novice you follow the way of life of the brothers and, if the other monks decide to accept you, you become a full brother. In some monasteries there are ranks of brothers like junior or senior. As a brother you take the vows specific to your community, which often include things like vows of chastity, obedience, poverty, and vows to pray the liturgy of the hours. These are similar to, but different in nature to the promises (not vows) diocesan priests make at their ordination.
After taking their vows, some (but not all) brothers begin pursuing ordination to the priesthood, thereby becoming a Father. This shouldn’t be seen necessarily as a promotion, just a different vocation, just like priesthood isn’t the same as married life. If you are ordained a priest, you are still a member of your Order, but it becomes possible for you to take on parish duties and even move up the hierarchy like Pope Francis. If you are made a bishop after your ordination, you no longer are bound to obey your abbot but you are part of your monastic community forever.
The brothers of each Order do very different things and I encourage you to research many different ones because you may be drawn to one kind of spirituality and not another. The Jesuits in particular are quite different from this way of life, and if I recall correctly they don’t actually become brothers at all.