In the Latin Church, there are seven clerical orders, all mentioned together  in the Historical record by Eusebius (260 AD) in the 43rd Chapter of the 6th Book of his "Church History." The lowest 5 are ecclesiastical in origin; the higher two are of divine origin. The seven orders are, in descending rank:
 
The 3 Major or Sacred Orders:

I. Priests

• Bishops -  the First Degree of the Priesthood
  Bishops have the greatest authority and jurisdiction (aside from Popes and Patriarchs), and have the powers to ordain men into the diaconate and priesthood, and to offer the Sacrament of Confirmation (this last power they can delegate to a priest), to dedicate churches and altars, to consecrate chalices and patens and bells, and to preside at the benediction of abbots. They are said to exercise the fullness of the priesthood. The symbol of this office is the mitre.
   
• Priests - the Second Degree of the Priesthood
  The duties and powers of the priest are to confect the Eucharist at the Mass; offer the Sacraments of Penance, Communion, and Unction; to preside at the Sacrament of Holy Matrimony; to solemnly baptize; to preach; to teach, guide, and sanctify his sheep. With ordination to the priesthood, a man has received the fullness of the Sacrament of Order. The symbols of this office are the stole, the chasuble, a paten with bread on it, and a Chalice filled with wine.

The duties of the deacon are to handle the sacred vessels, to be of service to the priests and Bishops (inside and outside of the liturgy), to read the Epistle and Gospel at the Mass, to be general stewards, and to serve the widows and orphans. This Order is Sacramental, and the first of the three divinely-instituted grades of the hierarchy of Orders, the others being the priesthood and the episcopate. The symbols of this office are the dalmatic, the stole (worn over the left shoulder, as opposed to around the neck as priests wear them, and under the dalmatic), and the Book of the Gospels.
 

The duties of a subdeacon are to serve the deacon at Mass; to prepare the bread, wine, and sacred vessels for the Sacrifice; to present the chalice and paten at the Offertory, and pour water into the wine for the Eucharist; to chant the Epistle; and to wash the sacred linens. This office is non-sacramental, but it is now that the vow of celibacy is taken. The symbols of this order are the empty Chalice and the paten, basin and towel, two little cruets, and the book of epistles.
 

IV. Acolytes

The duties of the acolyte are to light the Altar candles, carry the candles in procession, prepare the water and wine for the Mass, and assist the priest during the Mass The symbols of this order are the candle, the cruet, and a linen bag. (Note that altar boys are sometimes designated "acolytes" and fulfill the duties of the acolyte during the Mass.)

In the early Church, the duty of the exorcist was to cast out demons. Now that duty belongs to the priest alone, but this minor order is kept in traditional priestly societies nonetheless. The symbol for this order is the book containing the Rite of Exorcism.

The duty of the lector is to chant the Epistle when Mass is sung without a deacon and subdeacon. The symbol of this order is the Book of the Epistles.

The duties of the porter are to ring the bells, to open the church and sacristy, and to open the book for the priest. Most of these duties have passed to the laity, such as sacristans, etc., but in traditional priestly orders, this clerical order is kept as an office and stepping-stone toward the priesthood. The symbol for this order is keys.

A man who is to become a priest first receives the "tonsure" i.e., he is received into the clerical state by being given a surplice and having hair shorn away at the crown of the head (over the last 400 years or so, the hair-shearing has passed out of use due to Protestant persecutions). With the tonsure, he becomes a cleric, but still has not received the Sacrament of Orders.

The tonsured cleric is then ordained to each of the Orders above, one at a time, receiving the power of each office, and ascending up through the ranks until he is raised to the dignity of the priesthood, at which time he receives the fullness of the Sacrament of Holy Orders.

Those of the Minor Orders are not obligated to pray the Divine Office or bound to the rule of celibacy (but if they marry they lose their office); however, once the Major Orders are entered into, there is no going back, and from the level of deacon on up, the actual Sacrament of Orders is received. Deacons receive partial fruits of the Sacrament, priests receive the totality of the Sacrament, with only Bishops having more authority.

The Sacrament
 

Holy Orders is the Sacrament by which men become priests and are given a sacred power "sacra potestas" to act in total sacramental identification with Christ (i.e., to act in persona Christi) in order confect Christ's Body and offer it up to the Father at the Mass for the remission of sins; to forgive sins through the Sacrament of Penance; to solemnly baptize; to preside during the Sacrament of Holy Matrimony; to offer Unction to the dying; to preach; and to otherwise teach, guide, and sanctify their sheep. With -- and only with -- the permission of his Bishop, he may be delegated to offer the Sacrament of Confirmation, but to the Bishop alone is reserved the power to ordain other priests (though a priest may be delgated to ordain men to the sub-diaconate and the minor orders).

As in Baptism and Confirmation, the Sacrament of Holy Orders leaves an indelible mark on the soul of the recipient and can never be repeated once validly received; once a priest, always a priest (even if a priest is laicized and removed from his office, this mark remains).

As said, the minister of the Sacrament of Holy Orders is the Bishop, and the matter of the Sacrament is the imposition of hands, which takes place during the beautiful ceremony of ordination. 

The form of the Sacrament is the words:

Grant, we beseech Thee, Almighty Father, to these Thy servants, the dignity of the Priesthood; renew the spirit of holiness within them, so that they may hold from Thee, O God, the office of the second rank in Thy service and by the example of their behavior afford a pattern of holy living.

Latin: Da, quaesumus, omnipotens Pater, in hos famulos tuos presbyterii dignitatem. Innova in visceribus eorum spiritum sanctitatis, ut acceptum a te, Deus, secundi meriti munus obtineant; censuramque morum exemplo suae conversationis insinuent.

The recipient of the Sacrament must be a baptized, healthy, heterosexual male, at least 25 years of age, who has a vocation from God, a strong Catholic faith, intelligence, a good moral character, and a life marked by sanctity. He must be committed to living a celibate and chaste life, and to prayer (especially the Divine Office, which he is obligated to pray), and must have been properly formed in seminary.

Traditional priestly formation, lasts for 6 years and includes a thorough study of Latin, liturgy, liturgical chant, philosophy, Theology, Church History, moral Theology, dogmatic Theology, and Canon Law. During the first year, they receive the cassock; during the second, the tonsure; during the third and fourth, they are ascend through the minor orders; in the fifth, they are ordained to the sub-diaconate and then the diaconate; and after the sixth, they are ordained priests. The seminarian's days are heavily scheduled, much like a monk's, with daily Mass, the Divine Office, classes, private study, and community devotions.

 
The Rite of Ordination

(from the Catholic Encyclopdia):

All the candidates...present themselves in the church with tonsure and in clerical dress, carrying the vestments of the order to which they are to be raised, and lighted candles. They are all summoned by name, each candidate answering "Adsum". When a general ordination takes place the tonsure is given after the Introit or Kyrie, the minor orders after the Gloria, subdiaconate after the Collect, the diaconate after the Epistle, priesthood after Alleluia and Tract. After the Tract of the Mass the archdeacon summons all who are to receive the priesthood. The candidates, vested in amice, alb, girdle, stole, and maniple, with folded chasuble on left arm and a candle in their right hand, go forward and kneel around the bishop. The latter inquires of the archdeacon, who is here the representative of the Church as it were, whether the candidates are worthy to be admitted to the priesthood. The archdeacon answers in the affirmative and his testimony represents the testimony of fitness given in ancient times by the clergy and people. The bishop, then charging the congregation and insisting upon the reasons why "the Fathers decreed that the people also should be consulted", asks that, if anyone has anything to say to the prejudice of the candidates, he should come forward and state it.

The bishop then instructs and admonishes the candidates as to the duties of their new office. He kneels down in front of the altar; the ordinandi lay themselves prostrate on the carpet, and the Litany of the Saints is chanted or recited. On the conclusion of the Litany, all arise, the candidates come forward, and kneel in pairs before the bishop while he lays both hands on the head of each candidate in silence. The same is done by all priests who are present. Whilst bishop and priests keep their right hands extended, the former alone recites a prayer, inviting all to pray to God for a blessing on the candidates. After this follows the Collect and then the bishop says the Preface, towards the end of which occurs the prayer, "Grant, we beseech Thee etc." The bishop then with appropriate formulæ crosses the stole over the breast of each one and vests him with the chasuble. This is arranged to hang down in front but is folded behind. Though there is no mention of the stole in many of the most ancient Pontificals, there can be no doubt of its antiquity. The vesting with the chasuble is also very ancient and found already in Mabillon "Ord. VIII and IX." Afterwards the bishop recites a prayer calling down God's blessing on the newly-ordained. He then intones the "Veni Creator", and whilst it is being sung by the choir he anoints the hands of each with the oil of catechumens.

The bishop then hands to each the chalice, containing wine and water, with a paten and a host upon it. This rite, with its corresponding formula,.. [signifying] the power which has already been received, is not found in the oldest rituals and probably dates back not earlier than the ninth or tenth century. When the bishop has finished the Offertory of the Mass, he seats himself before the middle of the altar and each of those ordained make an offering to him of a lighted candle. The newly-ordained priests then repeat the Mass with him, all saying the words of consecration simultaneously. Before the Communion the bishop gives the kiss of peace to one of the newly-ordained. After the Communion the priests again approach the bishop and say the Apostle's Creed. The bishop laying his hands upon each says: "Receive ye the Holy Ghost, whose sins you shall forgive they are forgiven them; and whose sins you shall retain, they are retained." - The chasuble is then folded, the newly-ordained make a promise of obedience and having received the kiss of peace, return to their place.

It is customary for those who attend the priest's Ordination and/or first Mass to kiss the palms of his hands which have been consecrated by holy oils. Palm-kissing at either time results in an indulgence of 100 days under the usual conditions. An indulgence of 7 years, under the usual conditions, is received for piously attending a priest's first Mass; the indulgence is plenary if the one attending is related to the third degree to the newly-ordained priest.

Pray for vocations and encourage any priestly vocation your sons might have. Teach your children to have the utmost respect for priests, by word and your own behavior. Don't call a priest by his first name; he should be addressed as, for example, "Father Manzione" or simply "Father." He should enjoy the place of honor (barring the presence of higher ranking hierarchs) at social gatherings. Kiss his hands to show reverence for the Eucharist. Let your sons see that to be a priest is to answer God's highest calling, and that the fruits of the priesthood are His merciful gifts to us. In this way vocations are nurtured and God's people can continue to be nourished with the very Body of Christ.

And pray for priests, who give us so much.