The specific issue between Amalric and his wife Agnes of Courtenay might be suitable for a separate question, but briefly, Amalric and Agnes shared a great-great-grandfather, so according to the church they were too closely related to be legitimately married.

In the 12th century, spouses were not supposed to be related within “7 degrees” - i.e. if they were sixth cousins or closer. Unfortunately that made it virtually impossible for European aristocrats to find anyone to marry, since they were all related within the prohibited degrees! This was eventually changed in 1215 to 4 degrees, which made things a bit easier.

Of course most people ignored this anyway, and dealt with it later if it ever caused problems. The church could also grant a couple special permission to marry. For Amalric and Agnes it seems that the church objected to their marriage but no one bothered to do anything about it. He wasn’t supposed to become king, so why worry about? But then his older brother King Baldwin III died unexpectedly in 1162, without children, so Amalric was his only heir. The church and the nobles of Jerusalem decided that Amalric couldn’t become king if he was still married to Agnes, so the marriage was annulled, i.e. the marriage legally never existed. However, their children (the future King Baldwin IV and Queen Sibylla) were recognized as legitimate. How can children be legitimate if the marriage never existed? Well that’s a good question too, but it was legally possible even if didn’t make much logical or spiritual sense. Amalric eventually married the Byzantine princess Maria Komnene, and Agnes remarried as well.

Being too closely related (consanguinity) shouldn’t have prevented Amalric from becoming king, and the church could have easily granted permission, so why was everyone so opposed to it? There must have been political reasons. Those reasons are pretty hazy now, but the important thing is marriages could be annulled, or any obstacles could be ignored, depending on the circumstances.

Divorce and annulment are not the same. Divorce means two people were married, and they still are married in a spiritual sense, even if they don't live together. Today, we typically have another category, separation, which is distinct from divorce. But that’s really what “divortium” means in Latin, a separation. It’s often expressed as “separation from bed and table”, i.e. the couple no longer shares the same home and no longer acts like a married couple. So the terminology can be a bit confusing due to the slightly different modern meaning of divorce, where the couple is no longer legally married. In a medieval divorce they were just separated, but still legally married.

Medieval definitions of divorce were ultimately based on Roman law, where divorce was allowed for various reasons; for example if one spouse committed adultery, or was cruel, or became enslaved, etc. There was no such thing as secular and ecclesiastical law in Rome, but as the Empire became Christianized, a division between church and state also developed and ecclesiastical law (canon law) became its own separate thing. The church claimed jurisdiction over marriage, or at least the spiritual aspects of it. Canon law on marriage was also influenced by Biblical teachings, where, for example, the married couple becomes one body that cannot be unjoined. So even if a couple was physically separated, according to canon law they could not be spiritually separated.

For awhile the church didn’t really have a lot of power to enforce this. But in the 11th and 12th centuries, the reform movement within the church (starting roughly with Pope Gregory VII) centralized the church’s authority on the Pope in Rome, and tried to apply one uniform church doctrine and law throughout Europe, including laws on marriage.

As part of the reform movement, canon law began to be studied and collected, and issues like divorce were more clearly defined. The major collection of canon law in Amalric and Agnes’ time was Gratian’s Decretum. Gratian collected older canon law but also provided his own opinion sometimes. According to him, divorce could occur if one party hadn’t freely consented, if the marriage hadn’t been consummated, or if one party was impotent. In his view, consanguinity and adultery were not grounds for divorce. Even if divorce could occur, neither party would be allowed to remarry as long as one of them was still alive.

So this was essentially the church’s position in the mid-12th century when Amalric and Agnes were married. There was no legitimate reason for them to get divorced. But Gratian and other students of canon law agreed that the church could bend its own rules, so, as I mentioned, Amalric and Agnes could have easily been granted approval to stay married and there must have been political reasons that forced them apart. The church did bend the rules a little bit after their annulment, since their children were recognized as legitimate.

Hopefully someone with more knowledge of the early modern period can talk about Henry VIII and England, but politics were important there too - the Pope certainly could have granted a divorce or an annulment, but Catherine of Aragon was the aunt of Holy Roman Emperor Charles V, and Charles had captured Rome and taken the Pope hostage a few years earlier. So things were pretty complicated, to say the least.

Sources:

James A. Brundage, Medieval Canon Law (Taylor & Francis, 1995)

James A. Brundage, Law, Sex, and Christian Society in Medieval Europe (University of Chicago Press, 1990)

James A. Brundage, “Marriage law in the Latin Kingdom of Jerusalem”, in Outremer: Studies in the History of the Crusading Kingdom of Jerusalem (ad Izhak Ben-Zvi Institute, 1982)

Bernard Hamilton, “The titular nobility of the Latin East: the case of Agnes of Courtenay”, in Crusade and Settlement (University College Cardiff Press, 1985)