Was there a TV program run about inquisition recently?

Just a few hours ago there was another question on the inquisition on AskHistorians which I answered:

Was the Inquisition the incredibly ruthless and brutal torture and death machine it is recalled as or was it, albeit with questionable goals, a judicial branch of the church with very clear rules?

As mentioned in response to /u/zebogo I have written a number of posts on the subjects of medieval heresy and inquisition - it's my field of study. I'll link to them here because it covers a lot of the questions you ask and more. Feel free to ask followup questions in the respective posts or you can post them here.

How dangerous was being a heretic?

Was the inquisition as bad as it's made out to be?

How did one become an inquisitor for the Catholic Church?

What sort of tortures were allowed, and disallowed, during the inquisitions of the 12-14th centuries?

To what extent was torture criticized or objected to in the middle ages?

Sorcerers, Witches and Magicians - what did these people actually do?

How common were witch burnings in the middle ages?

Ok, I'm gonna break this up, because there's a lot of answers to each of your questions depending on when and where you are. Prepare yourself.

Innocent III and the Albigensian Crusade

I'm going to start this with the caveat that Innocent III, one of the big players in the start of the inquisitions, is my favorite pope, largely because of the massive impact he had on Catholic doctrine. If you don't know about the Fourth Lateran Council, you're missing out.

We'll set the stage in the early 12th century, when a section of what is today southern France became home to a particular flavor of Christianity called Catharism. This area of France was already problematic to the French crown for a number of reasons, not least because it was contested ground between the Count of Tolouse, ostensibly a French vassal, and the King of Aragon, one of several Spanish Catholic rulers. For various reasons involving their teachings (which unfortunately survive almost entirely in the writings of their opponents), the Cathars were also strongly opposed to the Catholic church. For almost two hundred years, the kings of France tried to squash the Cathars through political means (confiscating their property, taking holdings from Cathar-sympathizing castellans, etc.) but their attempts were largely unsuccessful.

Enter Innocent III. Already a strong opponent of heresy, he called upon the King of France to militarize. Using the Crusades against the Holy Land as precedent, the French gather an army and proceed to sack their way across the Count of Tolouse's land, butchering Cathars and Catholics alike. In many ways, this is the first Inquisition -- a military undertaking, sanctioned by the Catholic church, but run by local nobility. This is a pattern that most future Inquisitions will take; an uprising or politically distatestful group is identified by the local rulers, permission is obtained from the papacy, and the hammer drops.

In the wake of the Albigensian Crusade, by the 1230s, the county of Tolouse has been brought firmly under the French crown. The Count of Tolouse is posthumously excommunicated in 1222, marking the end of any real military resistance to the extermination of the Cathar population, and the consolidation of French authority over southern France.

Innocent III also, at Lateran IV, forbids trial by ordeal (the process commonly pictured when people think of "the Inquisition" of testing suspected criminals through fire, submersion, etc), instead forcing all trials in ecclesiastic courts (which was where heresy and other "inquisition" crimes were tried), to be determined by a system used in German common law of oath-taking and witnesses who could attest to a person's innocence. However, the best way around this restriction was to obtain a confession, or to move the trial out of church courts and into the authority of the local lord.

Which leads us to...

The Drummer of Niklashausen -- Heresy in the Holy Roman Empire

The year is 1476. A peasant named Hans Bohm, out in the fields with his sheep, has a vision (or rather, a series of visions) of the Virgin Mary. She tells him to burn his drum, and spread her word -- not the canonical teachings of the Catholic Church, however, but one of social equality. He does just that. People start flocking to the little village of Niklashausen to hear him preach, and what he says is pretty nuts. The clergy are corrupt, he tells them, and have no right to own land or have tenant farmers or collect taxes. Everything, the Virgin Mary tells him, belongs to everyone; that's how God made the world. Aren't we all Adam and Eve's children?

Suffice to say, this does NOT go over well. As masses of peasants descend on Niklashausen, the local landowners, with explicit approval of their Bishop, send spies in to listen to Bohm preach at folk gatherings. Having heard what he's saying, the council cuts off all entry to Niklashausen, but when Bohm retaliates by encouraging his flock to arm themselves, the Bishop sends men to abduct Bohm in the night. The peasants march on the castle where Bohm is being held, and have to be beaten back from the walls with cannons.

Bohm is tried for heresy and burned at the stake, but his crime is just as much a political one as it is theological. The church gets involved and abducts him not in some secretive "we must keep the faith pure by rooting out heretics" way, as the "inquisition" is pictured in popular culture, but as concerned landowners using a theological justification for their actions. Note especially that the Bishop doesn't just act immediately, either -- there is a semblance of a process of law. Under the law, he has to get witnesses who can give oaths to the fact that Bohm is committing treason and heresy. Until then, he can't act.

The Alhambra Decree and the Jews of Spain

"Ah," you say, "these Frenchmen and Germans are all well and good, but what about the Spanish Inquisition?"

The year is 1492. After centuries of war, the last Muslim stronghold in Spain has fallen. Ferdinand of Castille and Isabella of Aragon rule over a diverse kingdom, but to a Medieval ruler, diverse means fractured and in addition, the crown is pretty broke. However, imposing heavy levies on their (relatively small) Catholic population is both a bad idea politically and frowned upon by the Church.

To make a long story short, they sign The Alhambra Decree, which states that every Jew in Spain has two months to liquefy their property and leave the country. Every one, "along with their sons and daughters, menservants and maidservants, Jewish familiars, those who are great as well as the lesser folk, of whatever age they may be, and they shall not dare to return to those places, nor to reside in them, nor to live in any part of them, neither temporarily on the way to somewhere else nor in any other manner, under pain that if they do not perform and comply with this command and should be found in our said kingdom and lordships and should in any manner live in them, they incur the penalty of death and the confiscation of all their possessions by our Chamber of Finance."

There's a catch, though. The Jews are told to take everything they own when they go, "as long as they do not export gold or silver or coined money or other things prohibited by the laws of our kingdoms, excepting merchandise and things that are not prohibited". That all goes to the crown. In one stroke, the King and Queen have solidified their power, made the Catholic church really happy, and conjured up a massive supply of money. All that remains is to enforce the laws -- and by making Jewishness illegal as well as heretical, the crown and the church are able to use local political assets and legal systems to bypass the restrictions imposed by Lateran IV, as well as use the precedents established by rulers across Europe for dealing with heretical law-breakers.

TLDR

How did papal inquisitions work?

Inquisitions don't work the way you picture them, with sadistic monks extracting confessions from "heretics" under torture. They're a political tool, rooted in the European crusading tradition, that lets the church and state leverage each other's assets to consolidate religious and political power, and stamp out rebellion and heresy at the same time.

How did they suspect someone of heresy?

They sent someone in to listen to what you were saying (or sometimes, e.g. with the Cathars, they just assumed everyone in your town was).

How did the trial run (i.e. who was the judge, and who, if anyone, represented the accused, etc.)?

Under local law, for the most part. Find people who can swear oaths that the accused broke laws, and/or get them to confess.

And how did they determine what sentence is appropriate/how severe were the punishments?

Again, local law. Normally, it's a capital offense, though sometimes conversion's a way to get out of it.

And (last question) how did they carry out the sentencing (since priests aren't supposed to kill)?

The inquisition's not run by priests. It's sanctioned by the church, but it's run by local government.

Sources If you're interested in the Albigensian Crusade, Laurence Marvin's The Occitan War is a fantastically detailed overview of political and religious causes for the war for Tolouse.

Peasant Fires by Richard Wunderli is probably my favorite book to refer to on life in the the Holy Roman Empire. It's short, it's beautifully researched, and the story's so cool. My little retelling doesn't do it justice.

I'll be totally honest, the expulsion of the Jews of Spain is a little bit out of my area of expertise. In addition toThe Alhambra Decree itself, the best source I have for the expulsion is the well-sourced but not entirely scholarly Cartoon History of the Universe III. It's still a good read, but I'll admit I'm letting you all down here.

EDIT: Tweaked my TLDR.