This term makes an innate assumption that a particular argument, in the sense of a position or thought, is your basis. From the perspective that one argument is the standard (or basis), any argument that goes against the argument which is your point of reference would be a "counterargument."
The term is somewhat simplistic, (not to be confused with simple), since it implies that there are only two, opposing sides, when in a particular situation there may be many perspectives which interact in ways that are more complex than simple, direct opposition. Consider, for example, the evolutionary theories which stem from the writings of Charles Darwin, and how they are placed in opposition to concepts of creationism and certain types of Christianity. One perspective is to view evolutionary theory and creationist theory as being diametrically opposed and therefore irreconcilable, i.e. counterarguments to one another. However, some people consider each of the two theories as being competent in its own way, but that the areas they describe are so different as to be not in conflict.