The element of ethos deals with ethics, or common moral ground. Appealing to a given audience's sense of ethics in a persuasive essay is an effective way to convince them your argument is sound and necessary. For example, if you are writing about the necessity of taking action against global climate change, you could argue that it is unfair for the current generation to harm the environment for future generations. This appeals to an audience's sense of fairness and equity and helps to support your argument that action is necessary.
Pathos deals with an audience's sense of compassion for others. Using pathos is appealing to an audience's emotions and feelings in support of your argument. An example of pathos with regard to our global climate change argument could use anecdotal evidence. You could argue that polar bear populations are dwindling as icebergs melt and set them adrift in the ocean. Though the extinction of polar bears will not affect most people in a given audience, it will affect their sense of common compassion for living beings. Pathos can be a powerful element of an argument, but a good persuasive essay will keep it balanced with the other two rhetorical elements.
The use of logos in a persuasive essay appeals to an audience's sense of logic and reason. The use of statistics, scientific studies and other kinds of hard data keeps an argument grounded in a persuasive essay. With regard to our global climate change example, you might use global temperature records or scientific climate studies as logical proof that climate change is real and needs to be taken seriously. Logos is an essential part of a persuasive essay, and very often the foundation of any sound argument. Logos can lay the groundwork in your essay and then allow you to appeal to pathos and ethos.
A conclusion for your persuasive essay should summarize the most salient points of your argument and utilize ethos, pathos and logos. Depending on the discipline, audience and terms of your assignment, you will probably favor one rhetorical element over the other two. A legal brief will most likely favor ethos, a science paper will favor logos, and a literary essay will favor pathos. That said, a great persuasive essay will appeal to all three elements in some respect.