So the winter was the opportunity for two different categories of games: snow games and indoor games.

A popular snow game still played today is snow snake. Dan Jackson has a video explaining the game here. Snow snake was traditionally played by men and boys and was originally widely played around the Great Lakes. It could be played on the ice or in tracks made out of the snow. A smoothed stick with a knob on the end like a snake head is thrown along the track, and the person whose snake goes the farthest wins. You can see the game being played on a track at the Ojibwe Winter Games in this video. It's also commonly played among the Haudenosaunee such as this video of the game being played by Onondaga people. The length and width of the snake and the track vary among different tribes. Snow snake was documented by early white Americans in writing, such as Walter J. Hoffman who wrote about the game being played among the Ojibwe and Menominee in the 19th century. Snow snake had a practical use too, as the technique was sometimes used to hunt large game in icy conditions. Plains Indians sometimes used buffalo ribs instead of wooden poles.

Another popular snow game was snowshoe racing. Like snow snake, which originated from the practice of sliding sticks with carved messages on them across icy lakes, snowshoe racing was related to messengers. According to 18th and 19th century accounts, Ojibwe messengers, for example, could run 100 miles a day. Intense training was needed for messengers to reach this level of skill, and snowshoe racing was one way of doing that. Snowshoes were a highly developed form of travel among Indigenous North Americans; although some examples of snowshoes in Central Asia date back thousands of years too, the snowshoe as we know it today is an adaptation of Indigenous American styles. Snowshoes enabled people to continue holding races even in the winter time, keeping up their agility during the winter months. Snowshoe races are still held at the Ojibwe winter games, which you can see an example of here.

Winter was also a time for games to be played indoors. Especially popular were dice games. Gambling was a very important part of social life for many Indigenous Americans. Archaeological evidence of dice games goes back hundreds if not thousands of years. People from dispersed villages would come together for major festivals, where gambling over dice games and other events formed an important means of exchanging goods and even making marriage alliances. Important games were often preceded by ceremonial rites.

Women were particularly known for playing dice games. Sometimes men were not even allowed to participate, but other times the men would be cheering the women on and making bets too. Dice games were huge spectacles, accompanied by dancing and singing around the competitors. Here's an example of the dice game as played by the Potawatomi:

Among the Potawatomi, after a feast, a blanket was spread out on the floor, and the women divided into two teams and sat facing each other, each side in a semicircle. Any number of women could play, but there were only four prizes, usually lengths of cloth in various colors. The gaming equipment consisted of a wooden bowl and eight dice. Six of the dice were thin circular disks; one was carved in the form of a turtle, and one represented a horse's head. They were formerly made of buffalo rib, but horse ribs were common in later times. One surface of each die was colored blue or sometimes red, and the other was left white. The bowl was held with both hands and the dice were shaken to the far side of the bowl. Then the bowl was given one flip and set on the floor and the score was counted. [...]

The scoring varied according to each tribe, and each woman kept her own score using beans in front of her. Each woman shook until she missed twice and then passed the bowl in clockwise rotation. The first to score ten points won the game, and her prize-a piece of yard goods-was given to one of the men spectators, who in turn was obliged to reciprocate with a gift of equal value in the future.

Making dice pieces was a highly skilled craft which women perfected. Archaeological evidence shows that dice sets were prized, often stored alongside exotic trade goods like seashells and turquoise pendants. Men had their own versions of dice games too, often called the hand game. You can see an example of a dice game here and a hand game here.

Sources

"Recreation and Games", Wisconsin Indian Resource Project, Milwaukee Public Museum

"Agimag Gagwejikanidiwin", Ojibweg Bibooni-Ataadiiwin

"Gooniikaa-Ginebig Ataadiiwin", Ojibweg Bibooni-Ataadiiwin

Culin, Stewart. 1907. Games of the North American Indians. Twenty-Fourth Annual Report of the Bureau of American Ethnology to the Smithsonian Institution, 1902-1903. Washington, D.C.: Government Printing Office. [Please note, I included this source because the Ojibwe include it on Ojibweg Bibooni-Ataadiiwin as a recommended primary source for learning more about the games they discuss, but it is over 100 years old and reflects a very white perspective on the subject.]

Hall, Molly Allison, "Parowan Valley Gaming Pieces and Insights into Fremont Social Organization", unpublished MA thesis, Brigham Young University