Introduction to logic
Logic is the analysis of arguments. An argument is formed out of a set of premises and a conclusion. When writing out arguments it is common to number the premises and then separate them from the conclusion by a horizontal line. For example:
1. Mary is in the den.
2. John is in the library.
3. The den is a room separate from the library.
Mary and John are in separate rooms.
The discipline of logic has recently been invigorated by its merger with the discipline of mathematics. In 1854, George Boole wrote the book, "The Laws of Thought", in which he applied the methods of algebra to the study of logic. This marked the beginning of a revolution in the discipline. Now, the modern discipline of logic is incomplete without a background in mathematics.
The idea of "set" is central to the modern discipline of mathematics and logic. A set can contain an infinite number of members or elements. Members in a set may be related in some way but do not necessarily have to be related. For example, there could be a set whose members contain: President Bush, Democrats, brown desk, the number 3, and Scooby-Doo. One important rule in Logic in evaluating sets is called, “Axiom of Extensionality”. This rule states for any sets A and B, A=B if and only if A and B have exactly the same elements. The axiom of Extensionality basically states that order and repetitions in a set do not matter.
Logic can be divided into two main types : Deductive and Inductive.
Deductive Logic 
Deductive logic makes arguments by drawing conclusions directly based on the premises, such as the example argument above. These arguments tend to resemble geometrical proofs. This type of logic gives us no new information, but as long as logical fallacies (discussed below) are avoided, it can be counted on to be correct.
E.g. Aristotelian Logic (a.k.a. Categorical Logic, Descartian logic or Categorical Syllogism)
Inductive Logic 
Inductive logic involves a certain amount of probability, assumption and generalization, but can give new information as well. A common example of inductive logic is the assumption by most of the people on the planet that the sun will rise tomorrow morning because it has risen every morning that anyone can remember. That certainly doesn't mean that the sun has to rise in the morning, but it probably will.
Logical Fallacies 
There are two types of logical fallacies: formal and informal. Formal fallacies have to do with the structure of the argument, and informal fallacies have to do with the content. These often have Latin names with English nicknames.
Fallacies: Resources 
Wikipedia's list of formal fallacies.
Wikipedia's more exhaustive list of formal and informal fallacies, but this page is short on citation.
Fincher.org discusses both formal and informal fallacies.
Nizkor's list of fallacies, not broken into formal vs. informal categories.
The Fallacy Files is a very exhaustive list, but you'll want to know what you're looking for.