Sports, for the purpose of this post, are games that are played using one’s body as a major component. There are two types of sports: contact sports and non-contact sports. Contact sports are where one player makes physical contact with another player.
Sports are a bit more complex than games, so if your setting isn’t developed enough to have more than a small amount of leisure time, they will likely have very few sports.
Sports pretty much everyone has in every setting:
>Racing: including foot racing and animal racing (both human mounted and not)
>>In children, there will also be races in terms of “who can do their chores fastest” (these can continue into adulthood, but for the most part, will be done by children)
Sports than many settings would have:
>Racing: including swimming races
Other types of sports:
>Ball sports: the players may be in direct or indirect contact with the balls, there may be one or more balls, all players might be actively engaged in the sport, or they may take turns.
>>Balls can be many sizes and made from many materials, in setting without much technology, materials such as cork, wood, inflated animal bladders, and stitched leather may be used.
>Shooting: this can be be with any projectile weapon, usually some manner of gun or bow; this can include other forms of marksmanship including things like knife and spear throwing
>Hunting: this generally will not be a sport when a society still relies on hunting for food, as it’s just another chore in such a society
>Fishing: like hunting, this will generally not be a sport in a society that still relies on fishing
>Water sports: this can include simple things like various swimming types and diving, and can move into things like boating, sailing, water-skiing, and others
>Ice sports: this will usually start with simple skating, then racing, and from there other types can come out, such as figure skating or team sports
>Snow sports: as with ice sports, this will start with creating a device that allows them more mobility while traveling over snow
>Martial arts: both with and without weapons (marital/melee weapons, not projectiles)
Many sports are combinations of the above types. There are also other types that have not been mentioned, this is not a complete list.
Many sports will have displays of skills that are desired in their society. Societies that value teamwork will likely develop team sports first. Societies that fight often will develop their own fighting styles that will turn to sport when they are no longer immediately needed. Societies that value physical prowess will develop sports that showcase an individual’s raw power, usually involving things like lifting, jumping, and throwing.
Even if your characters don’t enjoy sports themselves, they likely will know someone who does. If sports don’t play a major role in your story, you only need to have a name for the sport and a vague idea of what it is in case it’s mentioned in passing.