Celibacy means swearing or deciding to avoid all forms of sexual contact, often as part of a religious tradition. (It should not be confused with asexuality, which is simply an absence of interest in sexual contact.) Celibacy of some sort is featured in many major religions, including Hinduism and the Roman Catholic Church. Celibacy is only a semi-permanent choice for Buddhist monks who can leave the monastic life and marry at any time. Unlike abstinence, which is an important and unsuccessful part of conservative sex education, celibacy is considered to be life long, in general, and a vow of celibacy prevents a person marrying. Roman Catholic Priests are usually required to be celibate for life, see exceptions below. Protestants are generally relaxed about people, including priests, ministers, pastors whatever marrying and having sex afterwards, the Shakers were an exception and required even married couples who converted to stop sex with each other.
The requirement of celibacy is what makes some Roman Catholic, Protestant, and Eastern Orthodox priests, monks, and nuns frustrated. If they keep their vow of celibacy they are not allowed to do anything with a partner, or even to themselves, ever.
The Bible encourages singleness Matthew, 19:12  (assuming eunuch in this verse is meant figuratively rather than literally), 1 Corinthians, 7:7-8 . The Catholic Church formalised this call into compulsory celibacy for the clergy, and celibate orders of monks and nuns. Protestants have not accepted the requirement of celibacy, but many think certain people are called by God to singleness. Liberals tend to think celibacy and being single too long is unhealthy, and missing out on an important part of life — although it may be the natural choice for some people with unusually low sexual appetites (asexuality). The Bible however sees singleness as a call, not just for those with little sexual drive, but also for some who have to fight against their sexual inclinations in order to fulfil this call — some think this is the "thorn in the flesh" Paul alludes to in 2 Corinthians, 12:7-10. 
The Roman Catholic Church normally insists on priests remaining celibate, although it makes an exception for converted priests from the Church of England and for the priests of the Eastern Catholic Churches. There is evidence this requirement was at least in part because the Church wanted to gain control of land possessed by the priests and bishops and did not want married priests bequeathing property to legitimate sons. 
The Catholic Church has two different types of laws.
- Laws which the Church believes come from God and can’t change. (For example the Roman Catholic hierarchy insists God does not want women priests though some Roman Catholics and many Protestants disagree),
- Laws which the Church believes come from human beings within the Church and which the Church can change. (For example the requirement for priestly celibacy).
Up till now, the church hierarchy has decided that carrying on with priestly celibacy is in the best interests of the Church as a whole, even though they have allowed exemptions from it in particular circumstances. The shortage of Roman Catholic priests is so severe that priests are not available to give the last rites to the dying.  For believers these rites are important because they comfort dying people and their relatives, removing the requirement for celibacy would help ease the shortage. If the RC’s were a democratic institution they would listen to the large numbers wanting change. The RC church is undemocratic and change is unlikely. 
Of course none of this prevents priests having illegitimate sons and daughters. Human nature being what it is and Roman Catholic opposition to contraception being what it is that happens often. The sons and daughters of priests typically grow up in families without a father while the father remains dedicated to the Church. The Roman Catholic Church opposes gay adoption because they claim Hypocritially that they want children to have mothers and fathers. If they really believe that's important they should allow their priests to marry and give their children a normal family life.
In the Eastern Orthodox Church, married men are permitted to be ordained as priests, but a single priest is not permitted to marry. In both the Catholic and Eastern Orthodox Churches, bishops must be unmarried — most commonly celibates who have never married, although the Eastern Orthodox Church permits widowers to become bishops.
Criticism of religious celibacy
"For whenever society attempts to restrict expression of the sexual drive more severely than the human constitution will stand, one or more of three things must occur. Either men will defy the taboos, or they will turn to perverted forms of sex, or they will develop psycho-neurotic symptoms, such as psychologically-caused illness, delusions, hallucinations and hysterical manifestations of various kinds. The stronger personalities defy the taboos: the weaker ones turn to indirect forms of expression. "- 
Critics of religious celibacy claim that it causes sexual disorders, including paedophilia; they see it as contributing to the many child sexual abuse scandals which have been a thorn in the side of the Catholic Church in recent years.  " While important to guard against and watch for, it must be remembered that most people who are celibate will not form deviant desires, and many religions outside of Christianity have a long history of celibacy without associated abuses of children, women, or each other.
The frustrated sex drive can be redirected in a wide range of different ways, some may be seen as constructive but others appear destructive. Many experts suggest celibacy is unnatural and can damage the personality of the celibate person. 
Can't resist temptation
Not all priests, monks and nuns have the will power to stay celibate. Some engage in what would be considered healthy relations with consenting adults. If caught, they are typically defrocked. Others, sadly, interfere with the healthy emotional development of children they have access to. Strangely enough, when caught, these perpetrators of a terrible crime tend to keep their jobs. And their bosses who played a part in the cover up get sinecures at the Vatican as a reward.
- Where the King James translation reads "and there be eunuchs, which have made themselves eunuchs for the kingdom of heaven's sake", the New International Version says "and there are those who choose to live like eunuchs for the sake of the kingdom of heaven".
- History of Celibacy in Catholicism: Catholic Priests, Celibacy, Marriage
- Priest shortage leaves faithful alone on sickbeds
- Celibacy debate rages, but change unlikely
- Catholics Priests Have Fathered Countless Children, Only To Abandon Them
- Mediaeval Sexual Behaviour
- Sexual Repression: The Malady That Considers Itself the Remedy
- These critics include at least one senior church figure, Hamburg Auxiliary Bishop Hans-Jochen Jaschke who was quoted saying "celibate lifestyle can attract people who have an abnormal sexuality and cannot integrate sexuality into their lives. Does Catholic celibacy contribute to child sex abuse?
- The Wages of Celibacy
- Is it even possible to live a celibate life?
- Cardinal Bernard Francis Law of the Boston Archdiocese, specifically.
- Here’s what a nun’s naughty bit looks like. It also features a monk with an erection. This one works as desktop background but leaves a space.
- Here’s a monk and a nun almost naked. This one is the right shape for a desktop background.
Neither are copyrighted.
Adapted from an out of date RationalWiki article