An Argumentative appeal is a tool used to persuade. Aristotle divided these appeals into three categories logos, ethos, and pathos. Not to be confused with rhetorical devices, these are far more general.
Ethos makes them listen
Logos makes them believe
Pathos makes them care (enough to act)
Logos
Logos is an appeal to logic. This appeal uses facts, charts, graphs, and logical reasoning to persuade. Logos is the most important argumentative appeal because it is convincing; Logos makes the audience believe a certain idea.
Example
"you won't find a deer on this road, I've been driving it for 60 years and have never seen a single one"
Because this person has gone such a long time without seeing a deer, it is likely that another person won't see one as well. The speaker can logically assume that not a lot of deer will appear.
Ethos
Ethos is an appeal to ethics. Ethos persuades the audience to listen depending on how the speaker presents him/herself. Ethos is quality of the speaker granted by the audience. IF the speaker presents him/herself as having certain qualifications (college education, workplace experience), good references (recommendations from other people, testimonials), or the use of facts, trusted sources, charts, etc in a speech/presentation, he or she has likely been granted good ethos by the audience and in return, they will listen to what the speaker has to say.
Example
This would be like a famous athlete endorsing or wearing a brand's shoe; because this famous athlete likes it, then other people will likely believe his opinion on the shoe because he has probably worn a lot of other shoes because of his profession. People will assume this person is a credible source if he performs well, works hard, has a lot of athletic achievements, etc. He will also appear as a more knowledgeable person about something like athletic wear (specifically shoes) since this is part of their everyday life and success.
Pathos
Pathos is an appeal to emotion. Pathos coincides with the purpose- the speaker/author will choose the best emotion to evoke that matches their purpose (if used correctly) which typically will make the audience care more about the topic or act upon what is said.
Example
"Don’t be the last person on the block to have their lawn treated – you don’t want to be the laughing stock of your community!"