games is a string of the form <year> <games>, or just <games>. 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.
univ applies only to the use of this template for Chinese Taipei at any Universiade games, to make sure that the Universiade flag for TPE shows.