(Disclaimer: my Jesuit wheelhouse is mostly 16th and 17th century, so I can't help much with post-papal-suppression-and-comeback Jesuit education in the 19th/20th centuries.)

In an Early Modern context, Jesuit schools and the Jesuit education style were indeed markedly different from their peers -- dedication to founding schools and a commitment to studying a specific syllabus of materials rather than more a scattershot approach. The idea, at least in theory, was that all Jesuits would be capable of teaching and all Jesuits would act as teachers at one point or another in their career. All this is in line with the Jesuit mission -- highly active rather than cloistered, and not constantly having to hustle to remain financially solvent as a mendicant order. Other religious orders like the Dominicans acted as educators as part of their social function; Jesuits made education (both youth education and higher ed) a priority almost by happy accident, rather than by design. (Teaching isn't altogether absent as an aim from early accounts of the Society and its founders, but it's not the priority it would become later in the centuries either.) By the end of the 16th century the Order recognized the position it was in with both the rising number of Jesuit schools/colleges and rising enrollment, and issued the Ratio Studiorum -- a formalization of the structure of schooling for Jesuits and by Jesuits, its texts, and its disciplines, such as rhetoric, grammar, philosophy, etc. Lots of disputation, lots of competition, lots and lots of Aristotle, lots and lots of Aquinas. It's not the same thing as an abstract statement of purpose but it is a prescription for an educational philosophy that proved especially potent and also especially attractive in the Early Modern era, and it was also an educational philosophy designed to shore up against the allurements of the Reformation. From the Ratio:

[The prefect of studies] shall not give permission to students of theology and philosophy to have books of any and every nature. They should be allowed only those which have been recommended by their instructors and approved by the rector. The theologians should have the Summa of St. Thomas, the philosophers Aristotle, and in addition some selected commentary which they may consult in their private study. All theologians should have the decrees of the Council of Trent and the Bible, and they should become familiar with them. The prefect should discuss with the rector the suitability of readings in Patristic literature. He shall likewise give both theologians and philosophers some work of classical literature and suggest that they set themselves a definite, though not too rigid a schedule for such reading.

(Though the Ratio also has plenty to say about not citing Jewish authors as if they hold too much authority of their own, so it's very much a product of the Counter-Reformation era and its anxieties about Judaism as well.)

On a youth level, schooling was seen as an occasion for ministry -- all education in the sciences and humanities was in effect religious education, and a means of preparing young men for civic life as well as a means of identifying potential future Jesuits from the ranks of their pupils. (Though it wasn't purely a talent search -- which young men became Jesuits and why had a pretty close connection to who they knew in the Society.) In mission work, the Society's commitment to study and language mastery got seriously put to the test, with mixed results -- if you're interested in the specific face of religious education in Japan for both laypeople and prospective Jesuits I can hook you up with some reading there.

Jesuits performed other public activities besides teaching -- you might find them as crisis responders in plague-stricken areas or as missionaries both within and outside Europe -- but central to their image and to their work was taking education very seriously while also being highly educated themselves. The Society put a high premium on all teachers being competent specifically in their areas of study, rather than treating all teachers as broadly qualified by virtue of being gentlemen, and one of the big draws of the Society in the eyes of prospective joiners was this public image of Jesuits as refined and pious but above all, erudite.

But we do know that Jesuit teachers sometimes fell short of the ideal, and not just because of high standards either -- we have records of complaints within the Society about specific teachers whose grasp of Latin was subpar or even worse, their grasp of vernacular languages. Claude Matthieu, rector of the Jesuit College at Tournon in 1566, had a few choice words to say about the Jesuits teaching there:

Gaspard Baston speaks French a little better, but he can't say three words of Latin without committing either a horrendous blunder, or a barbarism, or both at the same time, so that whenever I see a visitor go into his class I am completely mortified.

Or concerning another unlucky Jesuit at the same College:

Joseph Mazencourt knows very little and is also extremely troubled and restless.

Likewise, complaints and disciplinary proceedings concerning teachers who were competent lecturers but failed to maintain appropriate decorum with students, and so on. The Early Modern Jesuits weren't educational superheroes or unilaterally a bunch of brilliant polymaths, and their involvement in education has been subject to the same vulnerabilities, pitfalls, and scandals as other Catholic orders' educational ventures, but the Society of Jesus' commitment to rigor and position in time gave them a jump-start relative to their predecessors within Catholicism.

Other random features of Jesuit education: Jesuits pioneered the idea of the school vacation as a necessary break from study rather than just as a break for religious observance, and Jesuit schools frequently incorporated drama and dance. (So they had a hand in the invention of ballet.)

Some reading:

• Allan P. Farrell's English translation of the Ratio Studiorum

• "Active Leisure: The Body in Sixteenth-Century Jesuit Culture", Cristiano Casalini (on vacations, leisure, exercise, and the body in Jesuit education)

• "Vocational Crises and the Crisis in Vocations among Jesuits in France during the Sixteenth Century", A. Lynn Martin (re. the well-founded frustrations of Claude Matthieu)

• "The Culture of the Jesuit Teacher 1548-1773", Paul F. Grendler (this is a great piece for understanding how Early Modern Jesuits viewed their duty to teach and also wickedly funny)

I wish I could think of a single book on Jesuit pedagogy to recommend you but I'm drawing a blank right now -- I can hit the library this weekend and try to hunt one down.

Mm. I don't think it's historically valid to call either the Jesuit focus on education or the order's reputation for education excellence a "happy accident." /u/cdesmoulins points out some distinctive features of Jesuit education, but misses why they mattered, why they pitched the SJ towards a position as the ranking scholars and teachers in the Catholic Church despite the rich Franciscan and Dominican legacies of scholarship, despite the focus of new women's orders like the Ursulines on the invention of education for all girls as a concept.

Of course Iggy Lo wasn't sitting around in 1530 saying, "In 500 years we'll be running amazing universities with the occasional good basketball team." But the very rapid entry of the formal Society into education was the result of a series of steps arising out of the shortcomings they saw in 16th century education. Essentially, the teaching structures of Christianity had been built by the mendicant orders to fit a later medieval context and needs, which were not the needs of early modern Catholics. The Jesuits were early modern Catholics without the baggage of tradition, and made decisions to fit their own social and intellectual context.

To set the stage: in 1215, the Church doubled down on its commitment to pastor the laity. This meant, above all, getting them into Church twice a year, once to say confession and be absolved of sin in order to successfully accomplish the second, reception of the Eucharist on Easter. The goal of this was simple, but not easy--the quest for salvation. To reinforce the importance of the sacraments and provide the moral instruction to (ideally) behave less sinfully or (practically) know what sins people had to confess, the Church turned to preaching. Although sermons by parish priests rose in importance, a lot of the weight was shouldered by the new mendicant orders, the Franciscans and Dominicans.

This had two relevant implications for their philosophy and infrastructure of education, one internal and one external. Internally, that meant an education focused narrowly on preaching and conducting the sacraments especially confession. The orders set up their own colleges at universities, churning out the equivalent of today's MDiv rather than our stereotype of the doctores like Bonaventure or Thomas Aquinas. Externally, it meant a sharp divide between preacher and preached-to, and preaching/education occurred in a pastoral context and focused on moral instruction, on right behavior.

Like the Franciscans and Dominicans 300 years before, the early Jesuits coalesced around a core spirituality of imitating Christ spiritually within themselves and externally through the vita activa, the active life as opposed to a secluded contemporary one. Also like their earlier time, this ministry initially meant securing Christendom through preaching, now in the face of the Protestant challenge instead of a variety of perceived heretical groups. But there were a couple of other key differences. First, lay people as much as the Church magisterium wanted, demanded, good preaching. Second, the content they demanded in their preaching was different. Over the late Middle Ages, "Christianity" sort of...condensed or precipitated from an inherent, necessary element of the Latin universe like air or water, into, explicitly, "the Christian faith." It became a set of principles that people had to learn and assent to. That's not to say there was a "choice" yet, but rather that Christianity was now seen as a religion among (false) religions. While moral instruction of course comprised the foundation of the new preaching, there was a greater emphasis on "knowledge" and "knowing things" in a modern sense.

As was common practice, the early Jesuits established collegia, essentially college dorms, for their students going to earn their licenses to preach--at first, at universities just like everyone else. The early novices straddled two worlds. On one hand, they received a humanist education in the mode of the lay students training for jobs in bureaucracy, trade, law, an education wrapped in secular texts as well as religious ones and oriented towards this world as well as towards God. On the other, the existing pastoral instruction was antiquated--narrow, condescending, and out of touch with what the people they hoped to reach were demanding.

So the collegia, within the first decade or so of the SJ's formal incorporation, already started hosting lectures and lessons towards better preaching. The teachers themselves had humanist educations, and carried those principles in their teaching--including the value of that kind of broad-based, world-focused learning.

The shift to Christianity as a religion of knowledge rather than a way of life was part of the general increase in the importance of book learning, of literacy and factual knowledge. The late Middle Ages witness an explosion in urban grammar schools for artisan-class boys with no intention of being monks, priests, or even going to university. Book Knowledge and formal education was important to be successful in society more generally. In Protestant Strasbourg, several women's convents managed to stay open and resolutely Catholic by filling a newly-necessary social niche: they began running formal schools for outside students. Teresa of Avila's cloistered, monastic Discalced Carmelite order gets all the attention, but despite increasingly shrill voices mandating strict claustration (cloistering) for women, orders like the new Ursulines also made their mark as educators for lay girls and women. Education was important.

The Jesuit collegia, blending religious and secular education, began to attract lay boys and men as well. The Jesuits had zeal, fire, perspective, and the money to fund it. They also had an overseas, evangelizing mission--and they brought their ideas about adapting to the spiritual needs of their audience to their imperialist efforts. While other male orders, especially the Franciscans, were no slouches at early modern mission, the Jesuits took the leading role in a key philosophical approach: acculturation of Christianity to local perspectives. This meant, among other things, working for the establishment of an indigenous clergy as quickly as possible--which meant schools.

In the 16th century, overall, the Jesuits seized onto an education-based apostolate because they saw a need for it--they were early modern Catholics enmeshed in the same early modern religious concerns as the people they sought to reach. Their focus on humanist education and on educating lay men as well as Jesuits weren't just characteristics of Jesuit education but keys to its social success.

Thanks to both u/cdesmoulins and u/sunagainstgold for the truly fantastic responses. From someone who lives in a country where an impressive amount of private schools adhere to one or another religious doctrine, I always found the Jesuit´s pursuit to be a little different. Their teaching always seemed to be constructed upon a humanistic sensibility lacking in those efforts of other congregations.