Ethos is one of the three elements of rhetoric (the other two being logos and pathos). While pathos is an appeal to the emotions and logos is an appeal to logic, ethos is an appeal to ethics. Ethos specifically highlights the character of the orator.
In any oration or sermon, the speaker's credibility is reinforced when he can provide evidence that highlights (among other things) his flawless reputation and impeccable credentials. Basically, the speaker needs to show that he is qualified to argue his claims. In the sermon you reference, Jonathan Edwards uses a few appeals to ethos to reinforce the necessity of his listeners repenting and being born again (in a spiritual sense).
He quotes readily from the Bible, showing that he is well-versed in the Scriptures and therefore, qualified to dispense spiritual advice to his multitude of listeners. One of the verses he quotes is from John 3:18 "He that believeth not is condemned already." Edwards uses this verse to argue that every unconverted parishioner is in danger of the fires of Hell.
Throughout his sermon, Edwards reinforces the idea of an angry God presiding over the affairs of men. He uses various Bible verses to back up his claim. Edwards maintains that even an earthly king becomes angry when his subjects disobey him. According to Edwards, by way of the divine right of kings, a monarch has the right to expect perfect obedience from his subjects: rebellion in any form is considered to be treason against the king.
From this, Edwards argues that, if an earthly king has the right to expect obedience, then God himself (as a heavenly king) has the right to expect commensurate obedience in affairs of the soul. In fact, Edwards argues that God has paramount claim on the souls of all men.
All the Kings of the Earth before GOD are as Grasshoppers, they are nothing and less than nothing: Both their Love and their Hatred is to be despised. The Wrath of the great King of Kings is as much more terrible than their’s, as his Majesty is greater. Luke 12. 4,5. And I say unto you my Friends, be not afraid of them that kill the Body, and after that have no more that they can do: But I will forewarn you whom ye shall fear; fear him, which after he hath killed, hath Power to cast into Hell; yea I say unto you, fear him.
In keeping with his tendency of using his strong command of scripture to reinforce his claims, Edwards ends his sermon with a biblical allusion: "Haste and escape for your lives, look not behind you, escape to the Mountain, least you be consumed." This exhortation refers to the angel's warning to Lot and his family to flee before God destroys Sodom (Genesis Chapter 19).