Before we get to your question about Arthur Miller's The Crucible, let us first look at the definition of your chosen term (provided here by Carson-Newman University):
PATHOS (Greek, 'emotion'): In its rhetorical sense, pathos is a writer or speaker's attempt to inspire an emotional reaction in an audience—usually a deep feeling of suffering, but sometimes joy, pride, anger, humor, patriotism, or any of a dozen other emotions...In its critical sense, pathos signifies a scene or passage designed to evoke the feeling of pity or sympathetic sorrow in a reader or viewer.
It helps to think of Abigail Williams as the author of her very own play-within-a-play. She performs for the crowd using pathos to "evoke the feeling of pity or sympathetic sorrow." This occurs many times throughout the play; she feeds off the emotions of those around her in order to manipulate them. In the case of The Crucible, the most prevalent emotion is that of fear—fear of witchcraft and fear of accusation.
After being caught dancing in the forest late at night with a group of friends, Abigail confesses that there is nothing untoward going on: "Uncle, we did dance; let you tell them I confessed it. But they're speakin' of witchcraft; Betty's not witched." When she sees the adults around her getting worked up over the possibility of witchcraft, she then blames everything on "Tituba and Ruth." Knowing, as Betty explains a short time later, that she "drank blood" that night, she knows there might be grave repercussions for those actions. Thus, when Betty continues to mention it, she uses Pathos for the first time:
ABIGAIL: [...] Now look you. All of you. We danced. And Tituba conjured Ruth Putnam's dead sisters. And that is all. And mark this-let either of you breathe a word, or the edge of a word about the other things, and I will come to you in the black of some terrible night and I will bring a pointy reckoning that will shudder you. And you know I can do it. I can make you wish you had never seen the sun goes down!
Clearly shaken by everything that has happened thus far, Betty's emotional condition leads others to enter the room. This leads Abigail to claim that it started while they were praying, which only feeds their fears of witchcraft. Though this example does not directly answer your question, it does give some background into how Abigail learns to manipulate those around her using pathos.
After Abigail's advances toward John Proctor—a man with whom she was previously engaged in an affair—are rebuffed, she begins setting her sights on Elizabeth, his wife: "She is blackening my name in the village! She is telling lies about me! She is a cold sniveling woman and you bend to her!" Here, Abigail uses pathos to win John's affections. When that doesn't work, the begins to employ a much stronger plan.
While it is not shown in the play, Abigail's strongest use of pathos comes when she (presumably) stabs herself in the stomach with a needle. Ezekiel Cheever, a clerk of the court, explains what happened:
CHEEVER: The girl, the Williams girl, Abigail Williams, sir. She sat to dinner in Reverend Parris' house tonight, and without word nor warnin', she falls to the floor. Like a struck beast, he says, and screamed a scream that a bull would weep to hear. And he goes to save her, and stuck two inches in the flesh of the belly he draw a needle out. And demandin' of her how she come to be so stabbed, she... (To Proctor) testify it were your wife's familiar spirit pushed it in.
Now, most people would have a tough time believing Abigail would have done this to herself. Without knowledge of her affair with John or her attempt to get him to leave his wife, Abigail would seem like an innocent victim—even with the absurd claim that Elizabeth's "spirit" was the perpetrator. However, by this point in the play it has been shown that Abigail will go to great lengths to gain John's love, so it is safe (for the audience) to assume she stabbed herself and used the town's fear of witchcraft to try to get rid of Elizabeth.
Throughout the play, Abigail employs pathos in her behavior toward just about every other character in the play. For further study you can look into Abigail's behavior in court, where she employs pathos when she turns on Mary—who has testified on Elizabeth's innocence. Additionally, you can explore the beginning of the play, where she also turns on and blames Tituba for the events in the woods.
As the previous answer describes, pathos as a literary device has to do with an appeal to the emotions. In addition to Abigail's performative behavior, in which she appeals to the emotions of the present at Reverend Parris's house, she also creates a sort of narrative about Elizabeth Proctor that reflects on her statements and behaviors with others. She has two primary goals: to clear her name of accusation that she herself was trafficking with Satan in the woods and to win the affection of John Proctor. She thinks her hold over Proctor is much stronger than it is, and given her youth, Proctor feels somewhat guilty for his part in their affair in addition to the guilt he feels over cheating on Elizabeth. Abigail is aware of Proctor's complex emotions where she is concerned and goes out of her way to appeal to the passion she still thinks he holds for her. Unfortunately for Abigail, his sense of integrity and honor is stronger than his sexual feelings for her.
As Proctor begins to get an idea of just how far Abigail will go, in accusing Elizabeth of witchcraft, his previous pity for her turns to anger and outrage, and his rage unbalances him in his participation in the courtroom proceedings. It may be that Abigail also maneuvered this outcome: knowing Proctor has intense emotions, she may have considered how this might intensify his behavior and thereby make others judge him more harshly. By weakening Proctor, Abigail thought she could get him to turn to her, but by attacking Elizabeth, Abigail only succeed in strengthening Proctor's resolve to expose her lies and vindictiveness.
Pathos is a rhetorical device concerned with making an appeal to the emotions. Abigail Williams, in The Crucible, is a character whose wild accusations of witchcraft are based purely and solely on emotional foundations. This distinguishes her from Reverend Hale, among others, who tries to use logos or logical argument to persuade others that the accused are guilty.
While eating at Parris's house Abigail screams and falls to the floor, claiming she has been stabbed by Elizabeth's evil spirit using the doll that Mary Warren had given her. (She actually stabbed herself; this shows just how much she wants to destroy Elizabeth). She is effectively accusing Elizabeth of being a witch and using the seventeenth century equivalent of a voodoo doll to harm her.
The accusation is obviously ridiculous; the needle is used for sewing and has been stuck into the doll by Mary to keep it from being lost. However, Abigail is not trying to convince anyone on logical, rational grounds; she is appealing to the emotions. By this point in the play, emotions in Salem are running very high, so high, in fact, that in an atmosphere of near total hysteria people are prepared to believe just about any accusation of witchcraft no matter how demonstrably ridiculous.
