Using quotes when possible, explain why Abigail accuses Elizabeth of witchcraft in The Crucible.
The two previous answers are excellent, but I would like to add another motivating factor. Abby has constructed a lie that she is unable to extricate herself from...like a drowning person, she clings to whatever salvation she might find, even if it means dragging an innocent person (Elizabeth) down to save herself.
There is not much to recommend Abigail. She has already shown that she is ruthless in her desire to save her own skin. Consider her threatening of poor Betty and the other girls in Act 1:
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 saw Indians smash my dear parents' heads on the pillow next to mine and I have seen some reddish work done at night, and I can make you wish you had never seen the sun go down!
Then compare Abigail's selfishness to that of Proctor's eventual moral dignity in Act 3, speaking of Abigail: "She thinks to dance with me on my wife's grave! And well she might, for I thought of her softly. God help me, I lusted, and there is a promise in such sweat. But it is a whore's vengeance, and you must see it."
In addition to the other excellent answers provided here, I have another theory to offer. Perhaps the most obvious reason that Abigail accuses Elizabeth of witchcraft is that she can. In the prevailing climate of suspicion and accusation in Salem Village, the spiraling roster of names of the accused seems to enhance the excitement and titillation of the courtroom proceedings. Abigail and the other girls are named 'officials of the court' for their testimony, much of which consists of playacting in the form of demonic possession and childish acts of singsong repetition. Their social status is elevated for perhaps for the first time, since young girls who worked as servants did not enjoy much respect in the community, and their prospects for marriage were slim. The work was pure drudgery, and having the chance to escape it, as well as be looked up to as respected court officials, was no doubt a major motivating factor in the girls' continued public accusations.
Abigail has two motives for accusing Elizabeth. The first, most obvious reason, is to prevent John Proctor from exposing her. She knows that Proctor is aware of the fact that the girls are lying. Their exchange in Act I, Scene 2 proves this. If Abigail accuses Elizabeth, John has to be more careful about what he says. The court has accepted all of Abigail's testimony, and she can cause Elizabeth to be convicted easily, as is shown when she threatens Mary Warren from testifying against her.
The stronger motive, however, is that Abigail is in love with Johna and is convinced John loves her:
"You loved me, John Proctor, and whatever sin it is, you love me yet!"
If Elizabeth is convicted, Abigail would be the type of girl to assume that John would then be free to love her, and would, immediately. Abigail does not understand human nature, and does not see that John was looking for physical intimacy, but that he truly loves his wife.
Abigail accuses Elizabeth for both general and specific reasons. Abigail seems to enjoy exercising power (the general reason). She likes having a reason to threaten someone, and the ability to do so.
As far as a specific reason, Abigail was having an affair with John Proctor. Accusing Elizabeth is a way to get back at a rival, even destroy her.
Its her attempt to win John Proctor's love. She naively assumes that if Proctor's wife is dead, he will be free to continue the affair he originally had with her. However, this is where Abigail shows her "child"-like nature. She does not realize that John wants more than physical intimacy. If you read the last optional act, you'll see her confess that to John.
I would add that there is definitely a psychological factor to this. power is a new thing to this lowly maid, she no doubt uses this to make up for her previously worthless life. Now she can exact unjust revenge and take out her problems onto others. The phrase "oh ___ is just seeking attention" comes into play because Abigail did want attention, I would say that she was definitely not happy and depressed with her previous life.