Monopoly is a game that mimics business transactions. Rabbi Scheinberg, shlita, says that technically it is permitted, but playing a business-oriented game on Shabbat is not conducive to a proper attitude about Shabbat. Chess is discussed in the Shulchan Aruch. The Rama mentions it as a pastime that is permitted on Shabbat. Some of the Poskim note, though, that in order to differentiate between Shabbat and weekday, some people had special Chess sets made of Silver just for Shabbat play.
Chess clocks are another matter.
Chess clocks are two spring-wound clocks attached such that when one player finishes his move he pushes a button, stopping his clock and activating the clock of his opponent. Rabbi Scheinberg, shlita , rules that they cannot be used on Shabbat. When playing these games a Halachic consideration that should not be overlooked is the prohibition of Borer. Borer is prohibited when you separate something that you don't plan to use immediately, from a pile of other objects. With games that have many different pieces, this is something that can occur when cleaning up.