Many, many years ago, three elements of argumentation were formulated by ancient Greek orators. They were discussed in depth by Aristotle in his book On Rhetoric. Isocrates, Plato, Cicero, and others continued his discussion, mostly in terms of application and best usage. Those last two are important points because which element(s) is/are the best choice or most effective depends on the situation, medium, and audience.
Logos is basically logic and reasoning. It deals with verifiable data and syllogisms (premise-conclusion constructions). It is generally preferred in academic argumentation because the arguer needs to possess knowledge and understanding of the subject in order to make logos effective. That said, logos is not always the best choice for every situation, particularly because it takes time to construct a fully formed logos argument.
Logos is also prone to logical fallacies. Among the most common are the false premise (one element of the syllogism is demonstrably untrue), non sequitur (conclusion doesn’t follow from premises), straw man (false representation of the opposition’s argument), and the slippery slope (if A, then B, C, and D bad things follow). There are others, but these are among the most common.
Many people confuse the other two elements of argumentation with logical fallacies, erroneously. I think ethos gets mistaken because of the false authority fallacy (Person A is held as an authority when Person A has no expertise on the subject; commonly seen in advertising with celebrity endorsement). Logical fallacies are inherently things that pretend to be logical but are not (thus being false logic). Neither ethos nor pathos ever claim to be logical, therefore they are not logical fallacies.
Ethos is a false cognate. It looks like the English word ethics, which we do derive from the Greek ethos. But, there is more to the word. Ethos is a mix of the character of the speaker and the authority of the speaker. We use ethos regularly, often in conjunction with logos. Any time that we quote someone else on a subject to support our position, we’re using ethos (borrowing the other person’s authority). At the same time, by citing research, we establish our own authority on the subject.
Pathos is the source of the English words empathy, empathetic, and pathetic. Pathos is emotional appeals, which we see most often in advertising and politics. Generally, pathos is avoided in formal argumentation and academic argumentation because it demonstrates no knowledge of the subject. What pathos demonstrates is knowledge of the audience and what buzz words will get knee jerk, gut level reactions. It is still useful and effective in the right context, such as advertising. Pathos is also commonly used in speeches and other public speaking because it operates much faster than logos and requires less build up. Therefore, it is easier to follow—consider the average listening audience: will they follow an hour long monologue in which a complex logical argument is constructed or will they react better to a five minute speech that reaches the same conclusion through emotional appeals?