Tournaments were probably the most visible (and best recorded) of medieval "sports," although it's worth keeping in mind that our modern, Western conception of a sport is a very specific one, so it can be a bit anachronistic to think of a tournament or any competitive activity from a pre-modern period in exactly the same way we think of, e.g., baseball. With that in mind, some of what we know about pre-modern sport in Europe (with the caveat that this is a secondary area of interest for me, so I look forward to corrections and additions from experts here; I'm drawing heavily here from Allen Guttmann's Sports: The First Five Millennia):

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Tournaments—in particular early instances of the phenomenon—could be extremely violent, and might blur the lines of what we'd consider warfare and sports. In the 12th century the typical tournament was a melee—this might involve dozens of knights fighting and hoping to capture each other in exchange for ransom and/or fame. The "field of play" could be miles of territory that covered rivers, farms, vineyards, etc. There were no strict boundaries but, in general, towns provided some rough borders. There could also be some designated areas for rest or rearming oneself. Larger melees might involve several companies of knights "playing" under the lords they served in "real" warfare. Because this was meant for the knights themselves rather than for spectators, there were, initially, no provisions for spectators. Furthermore, a knight who came to watch might find himself inspired to join the event. Not surprisingly, as many (or more) people might die in a tournament as on an "actual" battlefield. An image from the Roman de Tristan, Harley MS 4389, c. 1300, gives an impression of what such a melee might look like (complete with an audience in this case): https://www.bl.uk/catalogues/illuminatedmanuscripts/ILLUMIN.ASP?Size=mid&IllID=28475

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There were efforts to regulate tournaments early on but they don't seem particularly effective: e.g., Richard I of England put forth regulations and gave authorization for them to take place at five particular sites (1194); Edward I issued Statua Armorum in 1292, which forbid spectators from arriving armed; and the Roman Catholic Church tried banning tournaments five times between 1130 and 1179 (unsuccessfully).

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The tournament became more elaborate (and regulated) between the 12th and 16th centuries. By the 16th century the joust had replaced the free-for-all madness of the melee. The tournament was also increasingly a place of theater and spectacle and pageantry, as much (or more) as it was a "sporting" event. It was also an option to demonstrate political power, either through the extravagance of the event or in the reenactment of victories on the "real" battlefield (real or imagined victories). Combatants began to use blunted weapons, a points system was developed for jousts, and ultimately the costuming and ceremony became even more important than the actual fighting in some cases. There's an example of what this might have looked like in MS Douce 383 from c. 1500: https://digital.bodleian.ox.ac.uk/inquire/Discover/Search/#/?p=c+0,t+,rsrs+0,rsps+10,fa+,so+ox%3Asort%5Easc,scids+,pid+de11a7fc-9a2a-4809-8a76-5a831f85fd04,vi+64851814-649a-44c5-8e44-1974a7f792ba

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A later manuscript, Knight games held by Emperor Frederick III & Emperor Maximilian I between 1489 and 1511 ( from c. 1550), is almost entirely focused on the armor and costuming of the participants, suggesting a further shift away from an interest in combat primarily.

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It's worth noting that participation in these tournaments was limited to male members of the nobility; some areas would occasionally open up a tournament to urban merchant classes (this seems to have been the case in Paris).

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Other sports: the sport available to all members of society was Medieval Football. Calling this a sport is a bit inaccurate ("sport" implies rules, boundaries, records, etc.) but it has enough elements to satisfy a loose definition of a sport or game. The game had few rules, no real distinction between players and spectators, could involve everyone in society (men, women, young, old, clergy, laity, rich, poor, etc.), and could be extremely violent. The game—which is also called "mob football" or "folk football" or "La Soule" in northern France—involves two opposing "teams" (often towns or parishes) trying to get a ball (likely an animal bladder) into the opponent's goal area (e.g., their church or a specific part of town). In general this was done just by handling the ball, rather than via kicking (although La Soule involved hands, feet, and sticks to hit the ball). As with tournaments, there were several attempts to ban this sport—the violence could easily lead to brawls and serious injuries—but efforts to ban it were ignored.

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An Old Scots poem from the 16th century gives an indication of how violent the game could be:

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Bruised muscles and broken bones,

Strife, discord and impoverished homes.

Stooped in old age then lameness too,

These are the beauties of football.

(available in original language here: https://en.wikipedia.org/wiki/The_Bewties_Of_The_Fute-ball

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Because some instances of the game took place around Mardi Gras there have been efforts to link its origin to fertility rites. In some cases the players might also be more specific: e.g., bachelors vs married men, married vs unmarried women, etc.

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By the Renaissance there were moves toward regulation of mob football, and we get closer to "modern" looking sports, but that's a different story.