Celibacy is the abstention from sexual activity. Usually, the word is used to describe abstention from all sexual activity for religious purposes.
The Catholic Church requires this of priests in the Latin Rite (the post-Vatican II "New Mass" and the most popular throughout the world) and Tridentine Rite (the pre-Vatican II "Latin Mass" used by some churches). Celibacy is not a requirement for priests of other rites (including Eastern Rite Catholics), or already-married Anglican priests who convert to Catholicism, though priests cannot marry after the death of their spouse (this has always been the case with deacons as well).
Celibacy is required of all bishops and archbishops (including their leader, the Pope). Monks and nuns also include celibacy in their vows upon joining a religious order.
While bishops were always celibate, as well as monks and nuns in religious orders, the modern celibacy requirements for priests were originally created due to legal issues regarding the inheritance of property. Over time, however, support for priestly celibacy became linked to the belief that it enables the clergy to be completely devoted to God and to their entire flock.
Although the word celibacy can also refer to the unmarried state, this usage is rare.
Outside of Catholicism
Most non-Catholic Christian denominations do not require priestly celibacy. Indeed, many Christians believe that while celibacy is a valuable spiritual discipline for those who are called to it, any requirement for priestly celibacy is contrary to the Bible.