How does Socrates use ethos, pathos, and logos in his arguments?
The previous educator response does a great job of answering this question. I'd like to add to her response by giving brief definitions of ethos, pathos and logos, along with a couple examples of their use by Socrates.
Ethos, pathos and logos are categories of argumentation or rhetorical appeal characterized by their use of different kinds of support for an argument. Ethos, from which we derive our English word, "ethics", is an appeal to the credibility or authority of the speaker. Ironically, Socrates often denies any knowledge of his own, claiming to ask questions because he is seeking a truth he has not yet acquired.
Pathos is an appeal to emotion for the sake of convincing an audience of the correctness of one's claim. The previous educator response indicates that Socrates deliberately avoids emotional appeals in his argumentation, which is absolutely correct. Socrates feels strongly that debates should be won by logic and sound reasoning rather than emotion or appeals to authority. For this reason, he primarily uses Logos, or logical arguments, to support his positions. This can be seen in all of Plato's renderings of Socrates's dialogues, one strong example being in the Menos when Socrates uses geometry and directed learning to illustrate the notion of innate knowledge.
Ethos: Socrates establishes atechne ethos by mention of his military service, representation of his tribe in the boule, and references to his acquaintances and associates in the audience. He establishes entechne ethos by refusal to use sophistic verbal techniques.
Logos: Socrates uses many different appeals to logos. For example, when he states that it is improbable that he could succeed in making people worse while so many others are invested in making people better, he is using the topos of greater and lesser.
Pathos: Socrates explicitly eschews appeals to pathos, especially in his peroration.