You know it's going to be a good story when it starts with Martin Luther getting mad.

Indulge Me

So the Protestant Reformation, the one that splintered the western Church that had lasted 1200 years? Was jump-started by, yup, cathedral funding.

The papacy needed money for construction on St. Peter's in Rome. The pope decided to update a long-standing method for the Church to collect money. So the Church offered a plenary indulgence to anyone who would donate. In non-medievalist terms, that means the Church was more or less letting people pay their future selves out of purgatory and into heaven.

Luther was not a fan of this for a whole host of reasons. Ninety-five of them, to be specific.

The 1515 St. Peter's case is extreme--indulgences weren't usually so nakedly materialistic. (Often they didn't involve money at all, but an indulgence in exchange for saying 505 Hail Mary prayers every night does not a cathedral build). One way or another, though, the Church's so called "treasury of merits" from which the spiritual benefits of indulgences came, was converted into the Church's treasury.

King Me

Obviously, most cathedrals didn't have the benefit of the papal court standing so firmly behind their construction. Here, patronage (i.e. donations) from one really wealthy individual or family could go a long way towards funding a project. Often this would work piecemeal--a stained glass window here, a spire there.

King Henry III spent 45,000 pounds of England's money to build a single extension on the church at Westminster Abbey. (For context, 13th century England had an annual revenue around 35,000 pounds. Per year. No, the cathedral arm wasn't built in a year, but sheesh.)

Dukes or other supremely rich nobles, and the wealthiest city patricians, also put forward money. I wouldn't call this "sponsorship" like of sports arenas today. You go to St. Theobald's church in Thann (Alsace), not the "Counts of Pfirts Cathedral."

Part of the motive was civic or regional pride--look at what my city can do. Some of it was (probably) genuine religious piety--wanting to build a tribute to God. And some of it was a more complex version of an indulgence campaign--often dedicating the donation to helping a loved one's way out of purgatory.

Taxation Without Representation

Not all, but many dioceses, monasteries, and even parishes in medieval Europe were quite wealthy. They could own huge tracts of land like any local noble, and tax resident peasants as heavily as necessary.

Parish churches would also collect tithes from regular parishioners, and preachers had plenty of stories about what would happen to people who held back their tithes. (Hint: you're gonna want those indulgences to buy your way out of purgatory. Trust me.)

Get Medieval

Local economy not strong enough to fund your late medieval cathedral? Turn to tourists pilgrims.

Medieval Christians believed very strongly in the miracle-working power of relics--body parts or other physical objects related to the saints, Mary, or Jesus that were left behind on earth. The fingers of Marie d'Oignies, the head of John the Baptist, Mary's breast milk. So Christians would go on pilgrimages to visit the shrines of relics, including at great churches. (Keep in mind that "pilgrimage" can be to the church on the far side of the city, or on the outskirts of a suburb. We're not just talking "Margery Kempe of King's Lynn, England goes to Jerusalem.")

...Of course, in exchange for overnight accommodation at the shrine or even for the privilege of being near the relic and seeing its reliquary (relic container), pilgrims were expected to donate a little money.

It wasn't just "my daughter went blind and I want St. Katherine to heal her" that drew people to relics and their host church, either. In the late Middle Ages especially, churches were empowered to offer indulgences (you knew this was coming) to pilgrims, too.

Where this part gets really crazy, though, is that religious leaders down to the local level were well aware of how lucrative a shrine/pilgrimage site could be. So they sprung up all over the place--and they'd be competing for pilgrimage traffic.

It was very advantageous for towns, especially, to offer special discounts to pilgrims in an attempt to lure them to the local shrine(s). Things like reduced or no-cost admission to the city for pilgrims, or badgering innkeepers to grant free lodging or a free meal to pilgrims. This would be especially useful if civic pride demanded a church to rival the best of them.

Hopefully I've illustrated the complexity and sheer medieval-ness of funding a medieval cathedral's construction. And FWIW, I highly recommend reading the very topical Pillars of the Earth by Ken Follett, which is probably my favorite fiction book. (Even if medieval guilds were not modern labor unions. Like at all. Oh well. You win some historical accuracy battles, you lose some.)

Allow me a correction to r/RileyRocksTacoSocks second paragraph. St. Genevieve, a pious fifth-century woman, commissioned the first church dedicated to St. Denis, the bishop of Paris martyred in c. 250 and made the patron saint of Paris and France. It was Suger, the abbot of St. Denis and counselor to King Louis VII, who, as part of a larger renovation, rebuilt the choir of St. Denis between 1140-44 with the novelties of large glass windows filling the walls and pointed arch ceilings, two of the hallmarks of what became disparagingly known in the Renaissance as the "Gothic style," i.e., barbarous like the Goths, not elegant like the classical architecture revival preferred in that era. Medieval people usually described it as "opus francigenum," the "French style." Suger's motive for the new style was both pious and practical. Certainly more light was a fitting metaphor for God--"the way, the truth, and the light"--(and incidentally, the light passing through glass without damaging it was a good symbol of Mary's Immaculate Conception). But enlarging the choir, where the relics of St. Denis were attracting hordes of pilgrims, allowed for easier access to the relics, more pilgrims, and more pilgrims' pious offerings. A very thorough and readable book that addresses your question about the motives and economics of cathedral-building is Robert A. Scott's The Gothic Enterprise, A Guide to Understanding the Medieval Cathedral (2005).

Chartres Cathedral may be an interesting example of "things aren't as rosy as they seem."

Most people know the story of the 1194 conflagration that burned down majority the late Romanesque structure, save for the west facade and the crypt. In the midst of the ruins, Chartres's most famous relic--the tunic of the Virgin Mary--was recovered unscathed. This was seen as a miraculous sign, and that Mary wanted a grander cathedral to house her relic (which drew in many pilgrims). The relic was actually taken on a fundraising tour in the region. Notable clerics and royalty also donated to the rebuilding; some also donated windows that decorate the finished cathedral, although the inscriptions in the window make it difficult to precisely identify, like "Cardinal Stephen". Construction was finished in a relatively quick amount of time for a cathedral, around 70 years.

However, as Jane Welch Williams describes in her book Bread, Wine, and Money, not everyone in the Chartres community backed the rebuilding. A local count began to regulate some of the tradesmen within the community (who weren't necessarily organized into guilds). However, many of these tradespeople sold their wares immediately around the vicinity of the cathedral (sometimes even inside, in the nave!), which was a sort of "free trade zone." This area was out of reach from taxation by the count, and the clergy was entitled to all taxes from sales in that zone. As you can imagine, this results in conflict between the count and the Chartres clergy, even leading to riots in 1258. Yet during this time, the tradesmen are, in effect, serfs to the clergy; this results in antagonism between clergy and tradesmen.

This seems to fly in the face of some of the iconography of the famous windows at Chartres. These windows, located in the lower side aisles, show various tradesmen (bakers, carpenters, blacksmiths, shoemakers, etc.) diligently working. In the past, scholars suggested that these windows were donations from each of the local trades groups, a pious offering to the building campaign. Yet Welch Williams argues that there's no documentary evidence suggesting any donation by these groups. Instead, the windows were a constructed representation by the clergy, in order to "control" these groups.

For more, see Jane Welch Williams, Bread, Wine, and Money: The Windows of the Trades at Chartres Cathedral (Chicago: U of Chicago Press, 1993)