There is a long history of formalized personal combat in European societies. It is generally traced back to traditions of trial by combat that were practiced by various Germanic tribes in the early middle ages. It was not a Roman/Latin custom originally, but eventually became one as it evolved by stages through different forms of chivalric and judicial combats, and long-running vendettas or blood feuds that were common throughout Feudal Europe.

The shift from legally recognized formal combats to unsanctioned or even outright illegal private duels happened as part of the shift to modernism at the end of the Middle Ages. The role of the knight as a cornerstone of social and military order was collapsing, and the relevance of the aristocracy as a "warrior" class was increasingly being drawn into question. Private duels afforded a way to reclaim that dying martial tradition, and assert the nobility's right to unilaterally inflict death when honour deemed it necessary.

Increasingly centralized governments did not think much of this vestige of medieval legal violence, and sought to control it, first by requiring such combats to be sanctioned, then by prohibiting private duels, and finally by vigorously prosecuting them. But despite these attempts to regulate private violence, duels remained popular, in no small part because they also served as a form of protest or demonstration of personal sovereignty by the aristocracy against attempts by royal houses and national governments to reign (sic) them in.

Another major shift in the sociology of duelling occurred around the time of the Industrial and French Revolutions. These two revolutions conspired to break down some of the class barriers that previously kept "commoners" out of the duelling tradition. Since men of all times and places seek upward social mobility, there was a pressure for successful members of the middle class to assert their "right" as gentlemen to engage in duels and lay claim to some of that aristocratic privilege. But without the traditional "martial" upbringing of real noblemen, they did not necessarily have good training in the use of a sword, so duelling was a bit of a risky endeavour (as, no doubt, the aristocracy intended).

Fortunately, pistol duelling was a thing, and was rising in popularity. Not a whole lot of skill was needed to pull a trigger. Even better for the not-particularly-murderous middle classes, it was possible for a pistol duel to run to satisfactory completion without anyone even getting hit. That was much better than sword duels, which were not felt to be satisfactory without blood being drawn (or worse, depending on the gravity of the initial offence, drawn from the trunk of the body). With a bit of luck, you might get through the whole affair without a scratch. Indeed, this was so attractive a proposition that the "luck" element of the encounter was actively cultivated by many participants, as discussed in another thread here. Pistols could be fired from long distances, they could be fired hastily without taking careful aim, they could even be fired into the air or ground if you wished to show contempt for the whole tradition. They were excellent weapons if you didn't really care much for the whole tradition and were just going through the motions for the sake of social standing. And if you did care for the tradition, well, you could always have your second negotiate for an uncomfortably close firing distance and a decent chance of having someone's head blown off. There was something for everyone!

The Duel in European History: Honour and the Reign of Aristocracy, V. G. Kiernan, Oxford University Press, 1988

The Duel, A History of Duelling, Robert Baldick, Spring Books, 1970