I suspect that game is either a Flash- or Java-based game, and neither come preinstalled on a Mac. You should be able to click on the "Missing plug-in" box to get information about what is required and where to download it from.
If Flash is required, you can download it from:
If Java is required, note that you should think very carefully before installing and enabling a Java web applet plug-in. Java has been a source of much grief (in the form of malware) lately. See Using Java in Mac OS X.