This topic has come up yet again in an e-mail discussion, so I decided to address it on my blog so that the next time it comes up, I can direct people to this page.
There is married clergy in the Catholic Church. There always has been. Both married and celibate men can be ordained to the diaconate; however, once ordained, a man cannot marry. If he freely choses to devote himself wholly to God through celibacy (cf. Matthew 19:12, 1 Corinthians 7:32-37) before ordination, he must remain celibate. If he was married before ordination and his wife dies after he is ordained, he cannot remarry but must remain celibate. It became the norm in the Latin Rite to only ordain celibate deacons to the presbyterate (priests). It became the norm in all of the rites of the Catholic Church to only ordain celibate presbyters to the episcopate (bishops).
No one is forbidden to marry in the Catholic Church; however, men must discern if God is calling them to celibacy or marriage before being ordained. After ordination, a man cannot marry. If a man is called by God to marriage, God will not call him to be a priest in the Latin Rite of the Catholic Church as long as his wife is living. As well, God will not call a married priest to be a bishop as long as his wife is living. God will not call a married deacon to be a Latin Rite priest or a married priest to be a bishop because God binds in heaven what the Church binds on earth (cf. Matthew 16:18, 18:18). Since the Holy Spirit inspired the Church to make these rules, the same Spirit will not confound them by calling someone to an office he can't legally be placed in.
The rule that married deacons can't become a priests in the Latin Rite, and the rule that a married priests can't become bishops are not doctrines and can change; although, it is very unlikely that this will ever happen. These rules were not always in place, but through the guidance of the Holy Spirit, they are in place now, and must be accepted.
I should point out that I said it was the "norm." Exceptions can be made. While the norm will always be followed for celibate clergy already in the Church, it is possible for married clergy from another church, such as the Anglican Church and some Protestant churches, to enter the Church and be ordained to the diaconate and then possibly to the presbyterate in the Latin Rite. I said, "possibly;" there are no guarantees, just as with any man discerning a call to ordination.
There is also another exception: a celibate priest can petition the Church to release him from his vow of celibacy, allowing him to marry. A release from this vow; however, comes with a serious consequence: he would not be allowed to normally exercise his priesthood. This is called laicizing a priest because the priest normally can no longer perform the functions proper to a priest, essentially making him a layman. He does, nonetheless, remain a priest because, like baptism, ordination cannot be erased. In case of a serious emergency, he is obligated to perform his priestly duty. As example of this would be hearing the confession of a dying man when it is impossible to get another priest there in time.
Aside from this exception with its serious consequence, ordained men are not allowed to marry. Married men are allowed to be ordained; although, it has become the norm in the Latin Rite to only ordain celibate deacons to the presbyterate and the norm throughout the Church to only ordain celibate presbyters to the episcopate.