Ethos, Pathos, and Logos are modes of persuasion used to convince audiences.  They are also referred to as the three artistic proofs (Aristotle coined the terms), and are all represented by Greek words. 
Ethos or the ethical appeal, means to convince an audience of the author’s credibility or character.
An author would use ethos to show to his audience that he is a credible source and is worth listening to. Ethos is the Greek word for “character.” The word “ethic” is derived from ethos.
Ethos can be developed by choosing language that is appropriate for the audience and topic (also means choosing proper level of vocabulary), making yourself sound fair or unbiased, introducing your expertise or pedigree, and by using correct grammar and syntax.
Pathos or the emotional appeal, means to persuade an audience by appealing to their emotions.
Authors use pathos to invoke sympathy from an audience; to make the audience feel what what the author wants them to feel. A common use of pathos would be to draw pity from an audience. Another use of pathos would be to inspire anger from an audience; perhaps in order to prompt action. Pathos is the Greek word for both “suffering” and “experience.” The words empathy and pathetic  are derived from pathos.
Pathos can be developed by using meaningful language, emotional tone, emotion evoking examples, stories of emotional events, and implied meanings. 
Logos or the appeal to logic, means to convince an audience by use of logic or reason.
To use logos would be to cite facts and statistics, historical and literal analogies, and citing certain authorities on a subject.Logos is the Greek word for “word,” however the true definition goes beyond that, and can be most closely described as “the word or that by which the inward thought is expressed, Lat. oratio; and,  the inward thought itself, Lat. Ratio. (1) The word “logic” is derived from logos.
Logos can be developed by using advanced, theoretical or abstract language, citing facts (very important), using historical and literal analogies, and by constructing logical arguments.
In order to persuade your audience, proper of Ethos, Pathos and Logos is necessary.
Examples of Ethos, Logos and Pathos:
Example of Ethos: 
"I will end this war in Iraq responsibly, and finish the fight against al Qaeda and the Taliban in Afghanistan. I will rebuild our military to meet future conflicts. But I will also renew the tough, direct diplomacy that can prevent Iran from obtaining nuclear weapons and curb Russian aggression. I will build new partnerships to defeat the threats of the 21st century: terrorism and nuclear proliferation; poverty and genocide; climate change and disease. And I will restore our moral standing, so that America is once again that last, best hope for all who are called to the cause of freedom, who long for lives of peace, and who yearn for a better future."
Democratic Presidential Candidate Acceptance Speech by Barack Obama. August 28th, 2008.
Example of Pathos:
"I am not unmindful that some of you have come here out of great trials and tribulations. Some of you have come fresh from narrow jail cells. And some of you have come from areas where your quest -- quest for freedom left you battered by the storms of persecution and staggered by the winds of police brutality. You have been the veterans of creative suffering. Continue to work with the faith that unearned suffering is redemptive. Go back to Mississippi, go back to Alabama, go back to South Carolina, go back to Georgia, go back to Louisiana, go back to the slums and ghettos of our northern cities, knowing that somehow this situation can and will be changed."
I Have a Dream by Martin Luther King Jr. August 28th, 1963.
Example of Logos:
"However, although private final demand, output, and employment have indeed been growing for more than a year, the pace of that growth recently appears somewhat less vigorous than we expected. Notably, since stabilizing in mid-2009, real household spending in the United States has grown in the range of 1 to 2 percent at annual rates, a relatively modest pace. Households' caution is understandable. Importantly, the painfully slow recovery in the labor market has restrained growth in labor income, raised uncertainty about job security and prospects, and damped confidence. Also, although consumer credit shows some signs of thawing, responses to our Senior Loan Officer Opinion Survey on Bank Lending Practices suggest that lending standards to households generally remain tight."
The Economic Outlook and Monetary Policy by Ben Bernanke. August 27th, 2010.