Where are pathos, logos, and ethos found in Animal Farm?

Ethos, pathos, and logos are all forms of rhetoric with the intention of persuasion.
Ethos is an appeal to a person's (or, in this case, animal's) sense of ethics. The speaker tries to solidify his position as a credible source or as an authority on the subject. A good example of ethos in Animal Farm is in Old Major's speech at the beginning of the novel:

"I have had a long life, I have had much time for thought as I lay alone in my stall, and I think I may say that I understand the nature of life on this earth as well as any animal now living.”

Here, Old Major establishes himself as an expert on life because of his experience and wisdom. He is credible because he has experienced so much. Therefore, he is an authority on life.
Pathos is an appeal to a person's emotions. In this form of rhetoric, the speaker tries to appeal to emotions or uses a convincing story to build credibility. This can be found in this excerpt:

“At the graveside, Snowball made a little speech, emphasizing the need for all animals to be ready to die for Animal Farm if need be.”

Here, Snowball is seen appealing to the animals' sense of loyalty, duty, and honor, trying to rally their spirits by uniting them under the cause of Animal Farm. This sense of emotional camaraderie is pathos.
And logos is an appeal to a person's sense of logic. In this form of rhetoric, it is common to find data: facts, statistics, graphs, and percentages. An example of this is in Chapter 8:

"On Sunday mornings Squealer, holding down a long strip of paper with his trotter, would read out to the animals lists or figures proving that the production of every class of foodstuff had increased by two hundred percent, three hundred percent, or five hundred percent, as the case might be.”

In this example, the other animals have some sense that the rhetorical appeal isn't quite right. It seems that maybe they are working harder than they were before. But when they are faced with the data that Squealer provides, they don't question it.

When used effectively and in combination (as in Old Major's speech), the three forms of rhetoric have the power to really bolster the credibility of the speaker.

Pathos, ethos and logos are the three pillars of persuasion. It is difficult to be persuasive in writing or speech without using all three.
Pathos is the appeal to emotions, ethos is the appeal to credibility or authority (do you believe the writer's or speaker's claims?) and logos is the appeal to logic, usually in the form of facts or statistics.
Although Animal Farm is a fable about animals, it uses all three forms of persuasion.
A chief example of pathos in this novel is Boxer, the strong, dedicated horse who is a true believer in the animal revolution and works harder than everyone else his whole life in honest, whole-hearted support of the dream. It is difficult not to be moved emotionally when he is betrayed in old age: rather than the retirement he has been promised, he is sold to the glue factory.
Orwell achieves ethos by making his animal characters behave like real, believable human beings and by basing his book on events in Stalinist Russia, such as show trials, which most people of his generation would have been familiar with. The book is also believable because we know some humans trick and betray others, so we can believe the pigs would act badly.
Orwell works hard to introduce logos into his tale, for example, by showing in some detail how the animals have to adapt human tools to their own four-legged state. This creates a world that accords with the laws of physics and thus feels real to the reader.