Overview

1. It demonstrates that you have fully researched your topic and are aware of all relevant arguments and issues, thus eliminating the possibility that your audience might be thinking, "Well, she/he only thinks that because he/she doesn't know about X."
2. It refutes or at least minimizes those arguments in opposition to your claim, thus giving your claim more weight.

The arguments that you will be countering fall into two categories:

1. Actual arguments that you have encountered in your research.
2. Anticipated arguments that you think might occur to a reader.

These arguments can be addressed in a variety of ways.  Below are some of the basic techniques.  

Type 1: The Facts in Evidence

1. The argument is demonstrably false: that is, that you have empirical evidence that refutes the argument.   Now, these are very rare, and most often occur with "common sense" arguments, those things that everybody knows. For example, everyone knows that there is more violent crime committed by teenagers now than there used to be.  However, actual empirical data demonstrates that youth violence has steadily declined since the late 1970's. Still, these arguments are rare.  More often, what you will find is:
2. There is more evidence that the argument is false than that it is true.  You will often find conflicting evidence.  One way to address arguments is to counter that the preponderance of evidence is against them.  Or:
3. The argument has not been proven to be true.  Even if you cannot prove that an argument is false, you can sometimes point out that no empirical evidence has actually been compiled to support the argument.  Again, the arguments vulnerable to this are often "common sense" arguments.

Type Two: That Does Not Compute

 I like to call these "your logic does not resemble our earth logic" arguments. These types of counterarguments focus primarily on the logic of the argument.  Variations include:

1. The logic of the argument itself is illogical.  One of the most common techniques is to invoke one of the logical fallacies, including:
   1. Slippery slope, in which the argument is made that a series of increasingly dire consequences will result from an action.  The key to this fallacy is that these consequences are exaggerated and unlikely.
   2. Ad hominem, in which an irrelevant attack on a person's character is used to distract from the issue. (Note: irrelevant. Pointing out that someone has lied in the past is not really irrelevant to a discussion of their truthfulness now.  Pointing out that someone cheated on her/his spouse in a discussion of a scientific study they conducted is certainly less relevant).
   3. False analogy, in which an analogy is drawn to a situation that is not similar enough to make the analogy valid.
   4. Post hoc (ergo propter hoc), or "after this, therefore because of this," in which correlation is assumed to mean causation.  These statements are often used to support that "X is bad for you."  For example, every single person who eats a tomato will eventually die.  (Think about it for a second).
   5. Hasty generalization, in which a general principle is drawn from two few examples.

   Note of Caution: The logical fallacies are good tools, but they're often overused and applied in overly simplistic ways.  As a teacher of mine used to say, some slopes are slippery. The key is knowing which ones really are and which ones people are just overreacting to.

2. There are logical holes in the argument.  This isn't quite the same thing as #1.  More often, it's a logical refutation of a factual assertion.  For example, death penalty advocates like to point out that there is not one actual proven case of an innocent person being executed in the U.S.  Factually, this is true.  However, one may counter that given the number of death row inmates who have been found innocent with technology we did not have even ten years ago, one might logically deduce that innocent people were executed before this technology became available.

Type Three: Yes, and?

1. The evidence is outdated: either science/technology has advanced to the point where the data is suspect, or society has changed such that the conclusions/arguments no longer apply.  Education arguments often fall prey to this, where arguments about how schools used to work/what students used to learn fail to account for changes in school populations and requirements.
2. The argument does not apply to the issue in question.  Just as one example, the Bill of Rights (particularly the First Amendment) is often invoked in situations in which the Constitution really does not apply - for example to a radio station deciding to fire a DJ.