Pathos, logos, and ethos are modes of rhetoric, which can be defined as the art of persuasion. In a piece of writing or, more commonly, a speech, an attempt is being made to persuade the reader or auditor that the relevant argument is one that they should endorse.
For example, a politician making a speech wants to try and persuade his or her audience to go along with what they're saying in the hope that they will vote for them. A defense lawyer in a criminal trial will use the art of persuasion to convince the jury that the prosecution has not proved the case against their client beyond a reasonable doubt. And so on.
There are three modes of rhetoric commonly used: pathos, logos, and ethos. Pathos is an appeal to the emotions and is the one most often used to persuade an audience of the validity of the speaker's argument.
A prime example of this can be found in Martin Luther King's famous "I have a dream" speech, where he refers to the lack of freedom for African-Americans a hundred years after the signing of the Emancipation Proclamation. This is an example of pathos in that King is reminding his audience of the hope—which is an emotion—that their ancestors once had that the end of slavery would mean equality between the races.
In the same speech, King also uses logos: an appeal to reason. This comes in the passage where he reminds his audience that there are many white people who are prepared to stand with African-Americans in their quest for justice. King wants to stir up emotions in his speech, but he also wants his audience to think, and this passage is a prime example of this.
Finally, King uses ethos, an appeal to an audience's ethical beliefs. He says he has a dream that, one day, his four little children will live in a country where they will be judged not by the color of their skin but by the content of their character. King is putting forward an ethical vision here, which he believes will correspond to that of his audience.