You might say the Catholic Church was the glue that held medieval Europe together. Unfortunately, that glue got a kick out of instigating campaigns of terror and death in the name of the big man on the throne in the sky. Executions ordered by popes or carried out in his name by Church authorities were not a civilized affair. Indeed, Vatican execution methods often involved long stretches of humiliation and torture in the build up to some of the most horrific deaths you can fathom. Heretical thoughts were serious business to Church leaders, so they dealt with them in the most serious ways they could. That seriousness brought about sadism and terror that's still a black mark against the intuition.
Who was persecuted? Jews, Muslims, accused witches, and really anyone who didn’t conform to the Church’s beliefs. Catholic Inquisition executions were often carried out by and at the discretion of friars or other minor religious authorities, acting on behalf of popes or monarchs. So, better not to piss off anyone associated with the Church, lest they elect to slice you to ribbons in the name of God. During this time you also had executions in Papal States, over which the Vatican held dominion.
Some free advice: if you end up time traveling to Europe between, say, the fall of Rome and 1700 or so, try not to piss off any Catholics. If you, it's possible one of these brutal execution methods awaits you.
Drawn and Quartered: Dragged by Horses and Ripped to Pieces
In the early 19th century, the Papal States (areas of Italy under direct papal authority), banned some of the more heinous methods of execution, though held on to drawing and quartering for crimes that were "especially loathsome." If you're thinking "That's an odd choice to hang onto while phasing out brutality," well, you're probably not alone, because this one's a doozy.
If you've never seen Braveheart, maybe you aren't intimately familiar with the process, so here are the basics: drawing and quartering has two parts. First, the condemned is drawn (dragged) by horses to the site of the execution. So that hurts like hell. Then the actual execution takes place, and methods are relatively diverse. One popular version of quartering involved tying each of the condemned’s limbs to difference horse and spurring them off. You can guess how that ended.
In another iteration, "hanged" was added to the sentence ("hanged, drawn, and quartered" is a common phrase in the torture and execution world). So there was the horse dragging, hanging, removal of the genitals, evisceration, decapitation, then the corpse was chopped to pieces (quartered).
Burning at the Stake: Business as Usual (Light 'Em Up)
Burning at the stake is kind of like the old school Catholic execution version of "Hey Jude." You don't see Paul McCartney and not get to hear that song, just like you don't get yourself through a inquisition without torching some people at the stake.
While burning at the stake was used for both sexes, it was the go-to for women because it was deemed too vile for them to be drawn and quartered. Joan of Arc, for instance, went out this way thanks to the Vatican (she was convicted of wearing men's clothes). Goes to show the lengths to which powerful (Catholic) men will go to avoid seeing the inside of a woman.
Men were burned at the stake, too. It happened to William Tyndale, who deigned translate the Bible into English for common people to read. Here's a helpful list of people were also lashed and roasted on papal authority for being heretics, and these ones were torched for witchcraft.
If you're wondering, burning at the stake is exactly what it sounds like - lashing the condemned to a wooden post (the stake), making a nice big bonfire, and letting it burn.
Boiled in Oil: More or Less Self-Explanatory
During the Roman Inquisition, it was a good idea to keep your head down and your ideas non-controversial. Unfortunately, civil law student Pomponio Algerio didn’t get that memo. His philosophical ideas against the church caught their attention, so they threw him in prison. After a year, he refused to change his views, so he was boiled alive in hot oil. It took him 15 minutes to die.
Water Torture: Forced Ingestion While Choking on Cloth
Water torture, also known as the water cure, was a common technique during the Spanish inquisition, used by those acting on Catholic authority to brutalize accused heretics and anyone else they didn't like. The accused had his or her mouth pried open with a bit of iron and a cloth stuffed into it. Then buckets of water were forced down the accused's throat for choking. The water also lodged the bit of cloth in the accused's throat, for a bit of extra choking fun.
In some cases, to be extra nasty, interrogators used boiling water or huge quantities of vinegar, expediting the process with a funnel. In at least one documented case, a man accused of demon possession and being a werewolf died of a distended stomach. Interrogators claimed the werewolf in him went a little nuts and drank so much water he died. Mm hmm. Sure.