Counterargument

A counterargument has two purposes and two parts.

Purpose one: to demonstrate to readers that you've conducted enough research to understand both sides of an argument.
Purpose two: by calling into question an argument against your position, it builds support for your own.

Part one: a specific argument against your own position.
Part two: a rebuttal which acknowledges or refutes or "counters" that argument.

Together, these two parts convey to the reader that you understand logic, are open to other views, and can employ basic rhetorical strategies in your writing.  Win, win, win.

Civic Stance essay: Conservatism

Thesis 
Conservatism is best for America because of its view on fewer business regulation and social programs, and for its support of limits on some private behaviors.

Counter argument for division #1

After paragraph arguing that fewer business regulations would help America, you could include:

Liberals, on the other hand, argue that these regulations, as Cooper and Peck MacDonald note, "keep powerful individuals or groups from trampling on the rights of the less powerful" (251). The problem with this view is that it interferes with an individual's right to decide what to do. [further explanation of this]

Or

However liberals argue that these regulations, as Cooper and Peck MacDonald report, "keep powerful individuals or groups from trampling on the rights of the less powerful" (251). This view position doesn't acknowledge that too often the federal government doesn't know when to stop. [further explanation of this]

Argumentative Discourse

Note the shift to argumentative discourse after the quote ("The problem with this view is that too often the federal government doesn't know when to stop"). Other sentence patterns that make this shift to argument include

The problem with ___ is that _____.

This ______ helps/hurts society/America/people because __________

The advantage/s of ________ is/are _______.

If _____ continues then _________.

Sentence patterns such as these literally force you to explain your point -- which is the goal in argumentative writing.