In order to understand what a fallacy is, one must understand what an argument is. Very briefly, an argument consists of one or more premises and one conclusion. A premise is a statement (a sentence that is either true or false) that is offered in support of the claim being made, which is the conclusion (which is also a sentence that is either true or false).
There are two main types of arguments: deductive and inductive. A deductive argument is an argument such that the premises provide (or appear to provide) complete support for the conclusion. An inductive argument is an argument such that the premises provide (or appear to provide) some degree of support (but less than complete support) for the conclusion. If the premises actually provide the required degree of support for the conclusion, then the argument is a good one. A good deductive argument is known as a valid argument and is such that if all its premises are true, then its conclusion must be true. If all the argument is valid and actually has all true premises, then it is known as a sound argument. If it is invalid or has one or more false premises, it will be unsound. A good inductive argument is known as a strong (or "cogent") inductive argument. It is such that if the premises are true, the conclusion is likely to be true.
A fallacy is, very generally, an error in reasoning. This differs from a factual error, which is simply being wrong about the facts. To be more specific, a fallacy is an "argument" in which the premises given for the conclusion do not provide the needed degree of support. A deductive fallacy is a deductive argument that is invalid (it is such that it could have all true premises and still have a false conclusion). An inductive fallacy is less formal than a deductive fallacy. They are simply "arguments" which appear to be inductive arguments, but the premises do not provided enough support for the conclusion. In such cases, even if the premises were true, the conclusion would not be more likely to be true.
Examples of Fallacies
Deductive Fallacy (formal error)
Premise 1: If Portland is the capital of Maine, then it is in Maine.
Premise 2: Portland is in Maine.
Conclusion: Portland is the capital of Maine.
(Portland is in Maine, but Augusta is the capital. Portland is the largest city in Maine, though.)
Inductive Fallacy (empirical error)
Premise 1: Having just arrived in Ohio, I saw a white squirrel.
Conclusion: All Ohio Squirrels are white.
(While there are many, many squirrels in Ohio, the white ones are very rare).
An informal fallacy is an attempt to persuade that obviously fails to demonstrate the truth of its conclusion, deriving its only plausibility from a misuse of ordinary language. Most scholars categorize informal fallacies as: (1) fallacies of relevance: appeal to ignorance, appeal to authority, ad hominem arguments, appeals to emotion, force, etc., irrelevant conclusions, and appeals to pity; (2) fallacies of presumption: accident, converse accident, false cause, begging the question, and complex question; (3) fallacies of ambiguity: equivocation, amphiboly, accent, composition, and division.