According to the Oxford English Dictionary, a fallacy is "a mistaken belief, especially one based on unsound arguments." Or, more specifically in the case of logic, a fallacy is "a failure in reasoning which renders an argument invalid."
There are two primary categories of fallacies: formal and informal fallacies. Formal fallacies are errors in the structure of the argument itself, and are all some form of a non sequitur. A list of formal fallacies can be found on Wikipedia here.
An informal fallacy is a fallacy that is content-based. Consider the argument:
- P1. If P, therefore Q.
- P2. P.
- C3. Therefore, Q.
There is no formal fallacy here, because the structure of the argument is sound. An informal fallacy, then, would be one where the content of the premises - that is, whatever P or Q is - means that the argument is wrong. For instance, take Bastiat illustrating the logical failures of socialist production: P is "The State provides Grain," and Q is "Only the State can provide Grain." Obviously, Q does not follow from P, even though the structure itself is proper.
There are many good resources that describe informal logical fallacies and give examples:
It's important to note that an informal fallacy doesn't necessarily mean that the conclusion itself is wrong; it just means that the reasoning used to get there was incorrect (see: "The Fallacy Fallacy").
A great chart:
Related: StatistFallacies (logical fallacies specific to statists)