The terms "formal" and "informal" can be deceptive at first glance, because people use the terms popularly to suggest modes of behavior and dress. The origin of the terms is the same for cultural modes as for argument, but the origin is obscured by the different venues---cultural and logical. The root word is "form." A cultural event that is formal dictates certain specific forms of dress and behavior. An argument that is formal conforms to a specific argumentative form.
Form in argument means structure. In a formal argument, the structure precedes the content. A syllogism is one example of a formal argument. It has two premises that share one category, which lead to a conclusion about an unshared category. For example: if premise 1 is that all humans are fallible and premise 2 is that Rebecca is a human, you might conclude that Rebecca is fallible. In a formal fallacy, the untruth or invalidity of a conclusion is related to failure to follow a structure, as opposed to inaccuracy of content. If premise 1 is that some humans are fallible, and the conclusion is that Rebecca is mortal, then there is a failure to follow the form, even though Rebecca is indeed mortal. The argument is invalid because the conclusion does not formally follow from the premises---a formal fallacy.
Formal logic is deductive; that is, the connection between the premises and the conclusion is seen as necessary and mathematically inescapable. Informal fallacies are non-deductive arguments that fail because of problems with the actual content, as opposed to the formal structure of the argument. A good deductive argument can prove its validity. Informal logic, however, does not provide mathematical certainty. It relies on probability, and the force of an informal or "inductive" argument is strengthened by a preponderance of evidence. "Most of the cats one observes are four-legged animals, therefore one can presume that a cat is a four-legged animal" is a sound inductive argument.
Informal fallacies are errors that arise from ignorance or selective representations of facts, misrepresentations of relationships and incorrect claims about how things work. Most informal fallacies are introduced by smuggling false or invalid claims into premises---like assuming that all women are illogical or that all microorganisms are harmful. Some informal fallacies rely on existing popular prejudices ("What she is suggesting is socialism!") or conclusions that simply do not necessarily follow from stated premises ("The brick is made of invisible atoms, therefore the brick must be invisible."). The more than 40 kinds of informal fallacy are listed widely online (see Resources).