The green-eyed monster is jealousy. The image comes from Iago, in Act III, Scene 3, when he warns Othello: "Beware, my lord, of jealousy." Jealousy, he says, is "the green-eyed monster which doth mock / The meat it feeds on", that is, a monster that mocks the one who feeds it.
Iago says this before presenting any proof. Othello answers that he would not live a life of jealousy, pursuing suspicion after suspicion, and that he wants "to see before doubting" and, once doubting, to have proof. Later in the same scene, he gives up that calm and demands ocular proof.
In Act III, Scene 4, Emilia takes up the idea: "Jealousy comes from jealousy." It is a monster "that begets itself and breeds itself". Desdemona asks the heavens to keep this monster away from Othello.
The word returns in other mouths. In Act I, Scene 3, Iago calls the plan he has just formed a "monstrous birth"; in Act IV, Scene 1, Othello says that "A hornèd man's a monster and a beast"; and in Act V, Scene 2, Montano calls the murder "Monstrous act!", and Lodovico calls Iago "this hellish monster".