The burning of people was not a common ocurrence, and the Spanish Inquisition had a surprisingly low convction rate, due in part to the procedural guarantees. The auto de fe was nothing but the spectacular final part of a very long judiciary procedure, and it was meant to be something public and impressive in order to instill fear into the hearts and minds of the heretics or apostates.

Let's skip the judiciary part, which can be rather ordinary, with attorneys, lawyers, evidence, witnesses, the whole lot, and get into what an auto de fe was like. Fernando de Valdés Salas, General Inquisitor, establishes this in the Instructions for the Inquisition:

Concluded all the procedures and the sentences having been dictated, the Inquisitors shall choose a holy day for the celebration of the auto de fe. The date will be notified to the priests, and to the municipal autorities, and in case there are, to the president and auditors of the Court of Justice, to invite them to the ceremony. The inquisitors will make sure the auto does not start too late so the execution of the sentenced people can be carried out in day light and without incidents.

Here we can see the relevance that publicity had, and the fact that it took a long time, that's why it should not start late. In a time prior to artificial lighting, it was important that public ceremonies were visible.

The preparation of the auto took a really long time, and because of that, it was a rather exceptional ocurrence, think of the logistics as something akin to a concert, but with fewer and worse means: the sambenitos, processional robes that the sentenced people wore to the gallows, needed to be made for each individual, and they were rather complex in design. These sambenitos were not burnt along the heretics, but hung in the parish to which the heretic had been ascribed, so everyone would be able to see them in the years to come, and the families of the heretics were publicly known, as they were barred from holding public office (see the Statutes of Blood Cleanliness by Cardinal Silíceo). The gallows had to be built ad hoc in the main square or in the close vicinity of the parish, as well as the stands for the audience. This requires some preparation and a lot of carpentry. In case one of the heretics had fled or died prior to the auto de fe, effigies (banners) had to be made so they would be executed in effigy.

A few days before the show, the town criers would announce the auto de fe, indicating the date, time, and place where it would happen, so the people would go see it.

The day before the auto de fe, the show starts with the "procession of the Green Cross". The green cross was a symbol of the Inquisition, along with the sword and the palm (quick side note here: in Spain there are many "plazas de la Cruz Verde" or squares of the green cross, named that way as the Inquisition was there). At two in the afternoon, the civil authorities, alongside the religious authorities would march in solemn procession carrying the green cross that would preside the auto.

The day of the auto, at dawn, the white cross procession would start. The procession was led by the public minister carrying a cross with some of the wood that would be used. Behind him, the clergy, then the effigies, then the reconciliados carrying large candles (reconciliados are those who had come back to the Church, abjurating their ideas, "de levi" or "de vehementi"), and the last ones were the people that would be burnt. When the procession made it to the square, the authorities would go to their places in the stands to watch the show. With everyone in place, the show starts.

First, a reputable and qualified preacher would pronounce a sermon defending the one true faith and condemning the heresies and the heretics. Then, the preacher would ask the unrepentant heretics to abjurate their heresies before they died. If they did, they were not burnt alive, but got strangled prior to being burnt.

Concluded the sermon, the individual sentences were read, with each sentenced person having to step forward. The reconciliados would then and there abjurate the heresies publicly, and swear not to commit the same errors again. With this part finished, religious hymns were sung, most notably the miserere me (God have mercy on me). This business finished, the burning of heretics finally arrives, but this is not the end. The reconciliados normally had pecuniary penalties and some sort of public penitence, which was carried out on the next day.

As I have said, autos de fe were a very rare thing, for they were incredibly costly (gallows, wood, sambenitos, salaries for al the workers...). The most famous autos de fe happened in the year 1559 in Valladolid and Seville. In those autos two very prominent protestant circles were eliminated, the group lead by doctor Agustín de Cazalla in Valladolid, and the group lead by Constantino Ponce de la Fuente in Seville. Ponce de la Fuente had been one of the Emperor's preachers and had worked as a censor for the Inquisition in Triana, that's why this auto was so prominent. In order to avoid these costly kerfuffles, the Inquisition always tried to get abjurations and reconciliations, with the imposition of monetary penalties, as cash was not always in excess in the Inquisition's coffers.

Sources:

Kamen, Henry (1999), "La Inquisición española". Barcelona: Crítica.

Torquemada, Tomás de, et alii (1657), "Instrucciones del oficio de la Santa Inquisición". Madrid: Diego Díaz de la Carrera. Digitised here.

Amézaga, Elías (1966), "Auto de fe en Valladolid". Bilbao: Gráficas Ellacuría