According to the Catholic Church, Christ gave authority to the disciples, which they in turn gave to their disciples, and so on.
Today, to become a Priest in the Catholic Church you must be called to the Sacrament of Holy Orders by the Holy Spirit.
Holy Orders is the sacrament through which the mission entrusted by Christ to his apostles continues to be exercised in the Church until the end of time: thus it is the sacrament of apostolic ministry. (CCC 1536)
Integration into ... [the Priesthood] was accomplished by ... a religious and liturgical act which was a consecration, a blessing or a sacrament. [This] sacramental act ... integrates a man into the order of bishops, presbyters, or deacons, and goes beyond a simple election, designation, delegation, or institution by the community, for it confers a gift of the Holy Spirit that permits the exercise of a "sacred power" which can come only from Christ himself through his Church. [I]t is a setting apart and an investiture by Christ himself for his Church. The laying on of hands by the bishop, with the consecratory prayer, constitutes the visible sign of this ordination. (CCC 1538)
In essence, a man senses the call of the Holy Spirit to Holy Orders (to become a priest) and consults his local Bishop who ensures this call is genuine, then makes a visible sign by laying on of his hands the called man.
Regarding similarities to the Levitical Priesthood "The liturgy of the Church ... sees in the priesthood of Aaron and the service of the Levites, as in the institution of the seventy elders, a prefiguring of the ordained ministry of the New Covenant." (CCC 1541) In other words, yes, the Catholic Church believes its priesthood is modeled after the Levitical Priesthood, or rather, the Catholic priesthood is the fulfillment that the Levitical Priesthood was modeled after.
In summary, the priesthood is more than a brotherhood of Christ. It is indeed the hands and feet of Christ, having Christ's authority to bless and curse, which was handed to the apostles by Christ himself and has continued in succession to this day in the priesthood of the Catholic Church.