In the Latin Church (the largest tradition within the Catholic Church), priests are usually chosen from among those men who live celibately. Celibacy means not only “not being married,” but rather specifically foregoing marriage for the sake of the kingdom of Heaven. Celibacy is a great gift both to the individual and to the whole Church because it means a closer imitation of Christ who himself was celibate. Jesus encouraged celibacy (Mt 19:12). St Paul did, too (1 Cor 7:7-8). The Eastern Catholic Churches typically choose priests from among both the celibate and the married. Even so, Bishops of the Eastern Churches are chosen from among the celibate.
Celibacy is an ancient tradition, going back even to the Apostles. Sure, some of the Apostles were probably married, but the Bible implies that they left their former way of life when Jesus called them (Lk 18:28-29). This does not mean that they abandoned their families, but rather that they no longer lived as married people but rather practiced continence (refraining from sexual activity). Even in the early centuries of the Church when married men were chosen to be priests, the expectation was that they would remain continent after their ordination. Celibacy soon became the norm, with married priests being the exception, and later on the Church decided that she would choose her priests only from among the celibate in the Latin tradition.
Continence is expected of priests because of the rich symbolism involved. Sexual activity is not bad, but it is a reality for this world (Mt 22:30). Priests are ministers of the things of Heaven, and so it is very fitting that they begin to live the life of Heaven even in this world, following the recommendation and example of Jesus himself. Priesthood is not just a job; it is an identity. Even in the Old Testament, the symbolism of refraining from sexual activity was very important for priestly acts (for example, 1 Sam 21:3-6). Again, this was not because sex was seen as something bad, but rather because priests mediated the presence of God to his people, dealing with eternal realities, while sex is a reality for this passing world. This is even more important in the New Covenant, where priests act in the very person of Jesus Christ. So, priests take the place of Christ, the Bridegroom of the Church. Celibacy for priests, then, means that they are radically available to the whole Church (as Christ is) without any one exclusive relationship taking priority. Celibacy also means that priests by their very way of life remind people to look forward to the heavenly wedding banquet. Priests have a spousal relationship to the Church, since they are configured to the person of the Church’s Bridegroom.
Priesthood in the Bible is strongly associated with fatherhood (e.g., Jdgs 17:10). The Passover meal, for example, had to be led by the father of the family (Ex 12:3). In the New Covenant, celibacy symbolizes that priests are fathers of the entire Christian people. Priests are not limited to a natural family, but they are spiritual fathers of the supernatural family of the Church. We even call our priests, “Father” for the same reason.