Q: Does the church ever use the word "divorce," or in every case is it always an "annulment?" Didn't Jesus allow for a spouse, in the case of a cheating spouse, to be granted a divorce, even calling it "a divorce?" (Indiana)
A: The church does use the term "divorce" in some contexts, such as when referring to the purely secular legal consequences of a civil divorce, e.g. child support arrangements or the division of assets. However, the Catholic Church sees divorce as being essentially a legal fiction, and not something that is truly possible on a spiritual, theological level.
Jesus teaches this clearly in Chapter 5 of St. Matthew's Gospel. Here, during the Sermon on the Mount, Jesus states: "It was also said, 'Whoever divorces his wife must give her a bill of divorce.' But I say to you, whoever divorces his wife (unless the marriage is unlawful) causes her to commit adultery, and whoever marries a divorced woman commits adultery" (Mt 5:31-32). Jesus goes on to make similar statements in Matthew 19.
So in other words, God created human marriage to be an absolutely permanent union -- to the extent that even if the spouses decide to divorce, this does not change the fact that they are ultimately still married to each other in reality.
Sometimes people have the mistaken idea that "annulments" are some kind of Catholic-approved divorce, but this is not at all accurate. The church's nullity process is based on Jesus' caveat of "unless the marriage is unlawful." That is, the church acknowledges the possibility, directly foreseen by Jesus, that there may have been some problem with one or both of the spouses, or with the union itself, at the time of the wedding, which was serious enough to prevent a true marriage from having taken place.
To give some common-sense examples, it would be "unlawful" for close relatives to marry, or to force a person to marry under threat of violence. Similarly, a person who for whatever reason lacks the mental capacity to consent to marriage is unable to fulfill the legal standard for marrying.
The church's nullity process aims to discern whether one of these "unlawful" elements was actually present and operative in a particular union. So when a declaration of nullity is granted, this is simply a formal acknowledgement that no true marriage occurred in the first place. This is different from the concept of "divorce," which supposes that a valid marriage could be ended. In fact, this is why the church does not officially use the term "annulment," which suggests actively rendering a valid union null; but rather "declaration of nullity," which is simply announcing what is already independently the case.
Yet even though the church does not believe in the true possibility of divorce, Catholics are permitted to separate in serious cases. As we read in Canon 1153, 1 of the Code of Canon Law: "A spouse who occasions grave danger of soul or body to the other or to the children, or otherwise makes the common life unduly difficult, provides the other spouse with a reason to leave."
Canon law also allows spouses to separate in the case of adultery. Although at the same time, canon law encourages the betrayed spouse to be "motivated by Christian charity and solicitous for the good of the family" and thus to forgive the offender (Canon 1152, 1).
Even in cases of what the canon law would call "separation with the [marriage] bond remaining," the church can condone a merely secular civil divorce as a sort of "last resort" for the sake of ensuring practical protections for those who might be vulnerable. As we read in the Catechism of the Catholic Church: "If civil divorce remains the only possible way of ensuring certain legal rights, the care of the children, or the protection of inheritance, it can be tolerated and does not constitute a moral offense" (CCC 2383).