games is a string of the form <year> Summer, <year> Winter, or just Summer or Winter. Wikipedia:WikiProject Olympics has created a full set of pages of the form "<country> at the <year> <Summer/Winter> Olympics" and the intent is that this template can be used from medal tables and event result tables to point to the right country page. Also, the correct historical flag is chosen based on this string, for countries that had different flags in the past.
athletes is the number of athletes that the nation contributed to the specific Olympics. It is an optional argument, and will render a small (#) after the nation's name.