Poskim disagree whether one may play games on Shabbat. Some say that since Shabbat is meant for Torah study, one may not play games, as that would be wasting time that could be used to study Torah. Accordingly, one may not play checkers, chess, backgammon, billiards, or any ball game, whether the games are played inside on the floor or outside on a paved area. And since one may not play these games, they are considered muktzeh as well (R. Aharon Sasson, cited in Birkei Yosef 338:1; Petaĥ Ha-devir ad loc. 4).
Others maintain that there is no prohibition on playing games on Shabbat, as long as nobody is playing for money (Rema 338:5; Ma’amar Mordechai). Indeed, some rabbis would play chess on Shabbat, as it is a game that requires thought and sharpens the mind (Shiltei Giborim).
In practice, it is appropriate for adults to be stringent and not play ball games, chess, and the like, both because it is prohibited according to some poskim and because one should not to get used to neglecting Torah study on Shabbat. Those who wish to be lenient have an opinion on which to rely (see SA 308:45; MA 338:5; MB ad loc. 21; Kaf Ha-ĥayim ad loc. 39). Children should also be trained to study Torah on Shabbat, but almost all poskim agree that one should not prevent them from playing games on Shabbat (as explained below in 24:7).
In contrast, sports that are a big deal, like soccer, basketball, baseball, and football, may not be played on Shabbat, because of the prohibition on weekday activities. It goes without saying that one may not play them on the court or field designated for them. Children may not play these sports either, because it is a weekday activity (below 24:9).