The cerebral game of chess has long captured rabbinic minds. References to the game can be found even in Rashi’s writings (Kesuvos 61b sv. de-mitalela). However, its status within Jewish law is complex and debated. Four areas in particular have sparked discussion among halakhic authorities — Shabbos, testimony, vows and idolatry.
You would be wrong to take for granted the permissibility of playing chess on Shabbos. The issues raised include: making sounds, conducting business, non-Shabbos behavior.
Apparently, on old chess boards, metal pieces that knocked into each other made musical sounds which might be considered forbidden on Shabbos. However, Shiltei Ha-Giborim (Rif, Eruvin 35b nos. 2-3) permits this because the players do not intend to make music with these sounds. The Rema (Shulchan Arukh, Orach Chaim 338:5) follows the Shiltei Ha-Giborim. Note that the Magen Avraham (ad loc., no. 8) confirms that the Rema is discussing chess (but requires using a special Shabbos set).
The Magen Avraham quotes from a R. A. Sasson who argues that playing chess is similar to conducting business. He seems to mean that because chess was often played for money, even when you omit the prize the game is still forbidden because of its usual practice. However, the Rema rules that chess is only forbidden when you play for money.
Many argue that chess is simply not appropriate for Shabbos. For example, the Shemiras Shabbos Ke-Hilkhasah (16:34) dismisses all arguments to forbid chess. However, earlier in the chapter (16:1), he says that this only applies to children. Adults should spend the day in spiritual–religious–pleasure. Similarly, R. Moshe Feinstein (Iggeros Moshe, Yoreh De’ah 3:15:2) rules that chess is not technically forbidden but should be avoided because of ve-dabeir davar, by which I think he means that the game is not in the spirit of Shabbos.1
Interestingly, the Chidah (Birkei Yosef, Orach Chaim 338:1) suggests that the great rabbis who have played chess, even during the week, must have done so in order to recover from depression. It was their strategy to heal and return to their Torah study. Therefore, their precedents cannot support a general permissive ruling.
J.D. Eisenstein (Otzar Dinim U-Minhagim, sv. shach) quotes literature surrounding a case of a man who vowed to stop playing chess but then regretted his vow. He asked for permission to annul his vow. The rabbi of Ancona replied that the questioner certainly took his vow to prevent wasting time. However, since chess is a game of skill and not chance, and it refreshes the spirit, he may annul his vow.
Eisenstein also quotes a discussion about professional chess players. Professional gamblers are barred from testifying in a Jewish court (Shulchan Arukh, Choshen Mishpat 34:16). Do chess players constitute gamblers? He quotes R. Yoel ben Nassan Finkerly of Alexandria who says that since chess is a game of skill and wisdom, it is not considered gambling. Chess players are generally highly intelligent and sophisticated. Therefore, a professional chess player is an acceptable witness.
Two years ago, we discussed whether owning a chessboard with a king that has a cross is permissible (link). While many say that Christianity is an acceptable religion for gentiles, all agree that it is forbidden for Jews. Therefore, we may not own Christian religious symbols. See there for an argument toward leniency.
It has since been brought to my attention that R. Asher Bush discusses this question in his Sho’el Bi-Shlomo (no. 60). He argues that since everyone knows that the chess piece is not connected to Christianity, there is no issue whatsoever and you may even display it in a place of honor in your home. He quotes R. Moshe Feinstein’s ruling (Iggeros Moshe, Yoreh De’ah 1:69) that you may sell stamps with a cross for two reasons–they aren’t made for religious purposes and people do not consider the stamps (religiously) important. R. Bush argues that the same logic applies to chess sets, particularly when the king is only one of many pieces.
 See also R. Eliezer Melamed, Peninei Halakhah, Shabbos 22:13 – link, in the harchavos, ibid., par. 2 and in this responsum: link