Shabbos Is a Day of Menucha
1. Learning Torah. Chazal tell us that Shabbosos and Yamim Tovim are for learning Torah (ירושלמי שבת ט''ו ג'). Workers busy with their jobs all week gather together on Shabbos to learn Torah (תנחומא, פסיקתא רבתי סי' כ''ג ). Talmidei chachamim who toil all week in Torah and enjoy food and drink on Shabbos should certainly set aside time on Shabbos to learn Torah (שם, ב''י סי' רפ''ח, של''ה שבת פ' נר מצוה אות נ''ג ).
2. Games. Therefore, gedolim throughout the generations wrote at length about people who spend their time on Shabbos on games. This is a problem of bitul Torah even during the week if it is at a time when one can learn, unless it is to relax a bit in order to gather more strength to learn Torah. One should certainly not spend his time on games and useless things on the holy Shabbos. He should learn Torah (שבט מוסר פרק מ''ב, ראשית חכמה שער הקדושה פי''ב, שו''ת מנחת יצחק ח''ג סי' ל''ג, שו''ת באר משה ח''ב סי' כ''ז ) and enjoy spiritual delights akin to Olam Haba, as whoever honors Shabbos by refraining from doing his own things is given a boundless inheritance (גמ' שבת דף קי''ח ע''א ).
3. Women, children. Women, who are not obligated to learn Torah, and children who are unable to learn Torah play all sorts of games on Shabbos, depending on their age. However, many games involve Shabbos issurim or at least potential Shabbos issurim without people realizing. Thus, we will attempt here to go through several types of common games, presenting the issues and the halachos.