Pathos, ethos, and logos are three ways of capturing an audience and providing evidence as to why they should support a writer's argument.
Pathos is a way to convey an emotional response by using past experience in the topic the writer is arguing about. It helps to establish a connection between the writer and the reader and create an emotion such as compassion or pity due to the writer's personal experience.
Ethos is a method of persuasion in which the writer attempts to persuade the audience by demonstrating his own credibility or authority. Like when political candidates say, "I approve this message," at the end of their campaign commercials, or when a professor writes "PhD" at the beginning of a syllabus to convey that he has a doctorate and is therefore trustworthy.
Logos is an appeal to logic. It is the consistency and clarity of an argument, as well as the logic of evidence and reason. Although arguments are also built on values and beliefs, they are also built on facts and evidence.
All of these are important tools to remember while writing argumentative papers or participating in a debate, but it is important to remember that not every argument needs all three elements. One argumentative essay could contain pathos and logos, but no ethos, and another could contain ethos and logos, but not need to contain pathos.
Pathos, ethos, and logos are all used in similar contexts. For example, if one is arguing with their roommate as to why they haven't cleaned up after their mess, the messy one could use pathos to make his roommate feel sorry for him by saying something like, "I'm sorry, I didn't clean my room because I've been so stressed lately because of finals, and I haven't felt motivated to do anything as of late." His roommate would then feel compassion on him for not cleaning, because perhaps they are going through the stress of college as well. The messy roommate could also use ethos by saying, "I read an a chapter in my psychology textbook stating that teenagers commonly have trouble adjusting to college life, and one of the ways they show it is by not cleaning up." This makes the messy roommate more credible by giving evidence to support his claim as to why he is messy, and makes his roommate trust him through his evidence. Logos could also be used in that situation, because it would be hard to deny the logic in the messy roommate's statement.
All of these methods are effective tools to use in an argument, or in an argumentative paper.