A bunch of very common misconceptions here we'll try to clear up; these questions are always great because they spur a whole lot of interesting discussions. First thing's first, we should differentiate between one-on-one sword dueling and battle tactics. They are two very different things.

Swordsmanship was an important component of the status of the landed elite, but the way it manifested itself varied greatly at different times, in different places, and for different individuals. I mostly draw from my knowledge of Italy between the 11th and 14th centuries, however these first paragraphs will give a general overview that if not complete for every time and place, broad lines can give you an idea of the kind of people we're dealing with.

Generally, it's safe to assume that landholders who drew their income from rents had the time and money to buy arms and practice with them, with the time and money available to them varying in proportion to their wealth and status. However, there are also very many cases of members of the upper middle class like notaries, doctors, lawyers, and moneychangers imitating the aristocracy (for both status and necessity) who also might not only own minor estates outside the cities where they lived, but also purchase and train with arms and armor. And this is by no means a universal truth: some aristocrats spent vast amounts of time in their city homes while neglecting their estates and had little to no familiarity with arms and armor, while some members of the upper bourgeoisie might only go into the city once or twice a week, and own and know very well how to use a complete set of arms and armor. As much as we like to think of ancient society as easily boxed, categorized, structured, and regimented, especially due to "feudalism," the truth is often much more nuanced.

That gives you an idea of the kind of person engaging in combat with swords. But with all that being said, it's also important to remember that swords are side-arms. They are certainly excellent status symbols: a sword is difficult to make, and has no purpose apart from battle (at least a spear can be used for hunting). Swords require lots of practice and training, meaning that those skilled at using them can signal to their peers they have lots of free time and disposable income. But when in battle, especially on horseback, a fighting man would almost always only use his sword in lieu of another weapon, at least until late in the gunpowder age when infantry became lightly armored and cavalry found it was useful to be armed with sabres. This is because, after the bronze age, battles were seldom swirling melees of one-on one duels: order and discipline was a crucial necessary condition for success. Infantry presenting a tightly packed wall of spears is nearly impossible to penetrate (or at least impossible to penetrate from the front. Which is generally why medieval tactics called for the cavalry to encircle and charge the infantry from behind. One of the most important advantages swiss pikemen had was their ability to advance and even turn while keeping a tight formation). Cavalry too, would repeatedly charge when lined up knee-to-nee armed with pole-arms, rather than engage for prolonged periods of time like you so often see in films and television (which is why the best way to combat cavalry was to bog them down with broken terrain or obstacles like woods, bushes, or mud). So generally, when a sword was drawn, an individual fighter has been isolated from his companions and/or is fighting in close quarters; and either way, can generally be said to be in a situation where something has gone wrong (with a prominent exception being made for cavalry in pursuit of a routing foe).

Of course, this role of the sword as an elite side-arm only strengthened its role as sort of freudian signal of status. Although popular media has an annoying habit of incorrectly depicting knights walking around doing their daily business in a full suit of armor, you can rest assured that a powerful lord and his retinue, when when walking around in "street clothes," would nonetheless prominently display their sheathed swords.

So, to (finally) get to the question, what did a sword duel actually look like? Two young men from rival families look at each other the wrong way in 13th century Verona and after some banter goes to far, draw their weapons; what do they do? We luckily know exactly what. Or rather, we have the best historical equivalent: a fencing guide from two centuries later that gives us some insight as to what one-on-one combat might have looked like. The guide in question, titled Flos Duellatum, is an early 15th-century treatise on swordsmanship written by one Fiore de Liberi. Fiore, mostly active in the later part of the 14th century, was a fencing master from a northeastern region of Italy called Friuli. The Friuli, a mostly agricultural and less-urbanized region compared to the Veneto immediately to the south, has sources attesting it had been home to multiple "Fencing Masters" for at least a century prior to Fiore and probably longer: "Magistro" Franceschino, "Magistro" Domenico Triestino, and Lucca Scarmitore, all mentioned to be living in and around Cividale, Friuli's most important city at the time. It would seem that with a strong landed aristocracy, fencing instructors were in high demand in Friuli. It wouldn't take much for their reputation to spread over all of northeast Italy, and Fiore was particularly popular, acting as an instructor for a score of important noblemen all over northern Italy, as well as to the monarchs of the Marquisate of Este.

Fiore de Liberi wrote his Flos duellatorum in the Este's chief city of Ferrara. The Duellatorum outlined instructions for several forms of combat, discussing different kinds of swords, quarterstaff, mace, wrestling, and even spear on horseback! It's imaginable that in one-on-one combat, a person would be able to use the knowledge contained in the manual holistically, for example moving from a sword parry to a wrestling grapple should the situation allow. Regarding swordsmanship specifically, you'd be surprised to know that Fiore's art is more similar to modern sabre fencing than any flailing and pirouetting you'll see in a film. In Fiore's manual (as well as others) a lot of emphasis is placed on posture, stance, footwork, and moving from one kind of guard to another. In northeast Italy especially, sword masters developed an almost maniacal attention to striking measure, time of the strike, and distance. But unfortunately, no one that I know of recorded specifically how long a medieval duel could take.

You might be interested in Medieval Warfare by Maurice Keen (Oxford University Press, 1999) which is a complete and wide-scoping, if cursory, overview of what medieval battles actually looked like. Or any of the innumerable translations of Italian fencing manuals published after the Flos duellatorum.