In the art of rhetoric, ethos consists of an appeal to ethics, an attempt to persuade an audience of the moral authority or credibility of the speaker. In short, it is a way for a speaker to let the audience know that he or she knows what they're talking about, which of course can only enhance the credibility of the message that they wish to convey.
In “A Talk to Teachers” James Baldwin uses ethos when he tells his audience that any attempts at reforming the American educational system—which is the subject of the lecture—will meet with “the most fantastic, the most brutal, and the most determined resistance.”
Baldwin is speaking from experience here. His own writings, which constituted a tireless attempt to challenge the numerous manifestations of racism in American society, were also met with fierce resistance, especially from those who benefitted from the status quo. Baldwin's experience as a writer and a polemicist gives him the authority to warn his audience of teachers of what they can expect by taking on and challenging the system.
Later on in the lecture, Baldwin resorts to another rhetorical technique: pathos. Pathos is an appeal to the emotions, a way of moving the heart. Baldwin does this by invoking the damaging effects of an unreformed, structurally racist education system on the African American child.
Such a child is liable to become schizophrenic, if only in the figurative sense of the word. On the one hand, he or she will be taught that America is the land of liberty, that anyone can become President, and so on. But on the other, he or she will also be taught that Black people have contributed nothing to civilization, and that their value as individuals is only measured by their devotion to white people.