A counterargument is the thoughtful response you give when you disagree with someone's ideas or claims. In criminal court, a defense attorney usually gives a counterargument to the prosecutor's accusations.
A counterargument is really just what it sounds like — an argument that runs counter to someone else's original argument. In this case, counter means "against" or "in response to," from the Latin root contra, "opposite, against, or in return." A true counterargument is well thought out, countering each idea with a set of reasons proving that the opposite is true.