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The Toulmin Model of Argument
In his work on logic and argument, The Uses of Argument, Stephen Toulmin indicates three major, necessary parts of an argument, along with three additional, optional parts. The three major parts are the claim, the support, and the warrants.

Claim: This is the main point, the thesis, the controlling idea. The claim may be directly stated (usually at the first of a text, but sometimes at the end, especially for effect) or the claim may be implied. You can find the claim by asking the question, "What is the author trying to prove?"

Support: These are the reasons given in support of the claim; they are also known as evidence, proof, data, arguments, or grounds. The support of a claim can come in the form of facts and statistics, expert opinion, examples, explanations, and logical reasoning. You can find the support by asking, "What does the author say to persuade the reader of the claim?"

Warrants: These are the assumptions or presuppositions underlying the argument. Warrants are generally accepted beliefs and values, common ways our culture or society views things; because they are so commonplace, warrants are almost always unstated and implied. The author and audience may either share these beliefs, or the author’s warrants may be in conflict with audience’s generally held beliefs and cultural norms and values. Warrants are important because they are the "common ground" of author and audience; shared warrants invite the audience to participate by unconsciously supplying part of the argument. Warrants are also important because they provide the underlying reasons linking the claim and the support. You can infer the warrants by asking, "What’s causing the author to say the things s/he does?" or "Where’s the author coming from?"

Here’s a visual representation and an example:

In this example, the claim that universities should reinstate affirmative action polices is supported by the reason that affirmative action provides equal access for all ethnic groups. It’s generally acknowledged by most Americans that equality of access is a basic American value.

There are three additional parts to Toulmin’s model of argument. Not every one of these is used in every argument, but only as need arises.

Qualifiers: Because argument is about probability and possibility, not about certainty, you should not use superlatives like all, every, absolutely or never, none, no one. Instead you may need to qualify (tone down) your claim with expressions like many, many times, some or rarely, few, possibly.

Rebuttal: When making an argument, you must take into consideration other conflicting viewpoints and deal with them fairly. You need to answer questions and objections raised in the minds of the audience; if you fail to do so, your own argument will be weakened and subject to attack and counter-argument. Sometimes rebuttal will be directed to opposing claims; other times rebuttal will be directed at alternative interpretations of evidence or new evidence.

Backing: Sometimes the warrant itself needs evidence to support it, to make it more believable, to further "back up" the argument.

These additional elements of argument may be added to our visual representation as follows:

Qualifier: If a university does not have a diverse student body
Claim: ...it should use affirmative action admissions policies.
Support: Affirmative action policies provide equal access to education for all ethnic groups.
Warrant: Equality of access is a basic American value.
Backing: Equality before the law is a fundamental right of all Americans.
Rebuttal: Affirmative action policies do not result in "reverse discrimination" because they are only part of a process that attempts to ensure fairness in college admissions.

Five Categories of Claims
Argumentative essays are based on a claim, which almost always falls into one of the five following categories.

1. Claims of fact. Is it real? Is it a fact? Did it really happen? Is it true? Does it exist?
Examples: Global warming is occurring. Women are just as effective as men in combat. Affirmative action undermines individual achievement. Immigrants are taking away jobs from Americans who need work.

2. Claims of definition. What is it? What is it like? How should it be classified? How can it be defined? How do we interpret it? Does its meaning shift in particular contexts?
Examples: Alcoholism is a disease, not a vice. We need to define the term family before we can talk about family values. Date rape is a violent crime.  The death penalty constitutes "cruel and unusual punishment."

3. Claims of cause. How did this happen? What caused it? What led up to this? What are its effects? What will this produce?
Examples: The introduction of the computer into university writing classes has enhanced student writing ability.  The popularity of the Internet has led to a rise in plagiarism amongst students.  The economic boom of the 1990s was due in large part to the skillful leadership of the executive branch. 

4. Claims of value. Is it good or bad? Beneficial or harmful? Moral or immoral? Who says so? What do these people value? What value system will be used to judge?
Examples: Doctor-assisted suicide is immoral. Violent computer games are detrimental to children’s social development. The Simpsons is not a bad show for young people to watch. Dancing is good, clean fun.

5. Claims of policy. What should we do? How are we to act? What policy should we take? What course of action should we take to solve this problem?
Examples: We should spend less on the prison systems and more on early intervention programs. Welfare programs should not be dismantled. The state of Oklahoma ought to begin to issue vouchers for parents to use to fund their children’s education. Every person in the United States should have access to federally-funded health insurance.

Just about any given topic can lend itself to be stated as one of the five types of claims. For example, the topic of gun control could be approached from any of the five different types of claims:

Claim of Fact: There are serious restrictions on our Constitutional right to bear arms. (This essay will give facts, examples, and statistics relating to laws and policies that restrict the sale and use of firearms.)

Claim of Definition: Laws governing the sale of firearms such as assault weapons and handguns do not constitute an infringement on our right to bear arms. (This essay will focus on the Bill of Rights and its clause about the right to bear arms. It will argue for a particular definition that excludes the writing of laws that relate to ownership of firearms.)

Claim of Cause: Tougher laws governing the sale of handguns would mean a decrease in the number of homicides each year. (This essay will seek to establish a link between difficulty in obtaining a handgun and a drop in the homicide rate. It will use statistics, facts, and analogies from other places where similar things have been done.)

Claim of Value: The right to bear arms is still an important civil right in the United States. (This essay will appeal to people’s sense of the value of gun ownership. It will probably appeal to authorities, such as the Constitution, to history, and to long-held customs.)

Claim of Policy: The sale of assault weapons in the United States should be banned. (This essay will use a variety of motivational appeals and value proofs, analogies, facts and statistics, cause and effect arguments, and appeals to authorities to prove that this is a favorable course of action.)