Counterargument: As the term itself implies, a counterargument is a response or refutation of an argument. For example, politicians and citizens might argue to return refugees to their countries of origin rather than granting them asylum in the United States. Depending on how this argument was made, those in favor of a more open asylum system would be ready with strong counterarguments: denying refugees aid is immoral; arguments about closed borders as a "national security" measure rely on racist sentiment rather than actual evidence of any real security threat.
So then, audiences make counterarguments, but thinking in terms of counterarguments is also a powerful strategy for writers. The writer we imagine in this refugee-controversy scenario would likely anticipate the economic and security concerns of politicians and look for ways to respond. Or to use another example, a student researching causality between violence in the media and gun crime would do well to carefully examine the many arguments and counterarguments circulating in the public sphere and among researchers in fields of study including sociology, criminology, psychology, media studies, and cultural studies.