"Order" comes from the Latin word "ordo", which can mean "rank", "class", "grade"—things of that sort. The Catechism, in its discussion of the Sacrament of Holy Orders, says:
The word order [sic, sc. ordo] in Roman antiquity designated an established civil body, especially a governing body. Ordinatio means incorporation into an ordo. In the Church there are established bodies which Tradition, not without a basis in Sacred Scripture, has since ancient times called taxeis (Greek) or ordines. And so the liturgy speaks of the ordo episcoporum, the ordo presbyterorum, the ordo diaconorum [respectively the "order of bishops", "order of priests", "order of deacons"]. Other groups also receive this name of ordo: catechumens, virgins, spouses, widows,....
Integration into one of these bodies in the Church was accomplished by a rite called ordinatio, a religious and liturgical act which was a consecration, a blessing or a sacrament. Today the word "ordination" is reserved for the sacramental act which 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" (sacra potestas) which can come only from Christ himself through his Church. Ordination is also called consecratio, for it 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.
(Catechism of the Catholic Church, paragraphs 1537–38)
So an "order" generally speaking is simply one of the groups the Church recognizes as having a particular purpose or charism; and "order" in respect of the clerical hierarchy means one of the three groups set apart for the ministerial priesthood: bishops, priests, deacons.