In the scene in which Macduff discovers the bloody corpse of King Duncan, the Porter, still suffering the effects of a night of drinking, pretends that he is the gatekeeper of hell. Among the sinners that he pretends to welcome into hell is an "equivocator, that could swear in both the scales against either scale" (2.3.8-9).
This passage is often considered to be a reference to Henry Garnet, a Jesuit of Shakespeare's time who wrote a "A Treatise of Equivocation." He wrote the "Treatise" in order to tell other Catholics how to deal with dangerous questions from Protestant inquisitors. If the Catholics admitted that they were Catholics, they would be in serious trouble with the Protestants. On the other hand, it was a sin against God to lie under oath. The solution to the problem, Garnet said, was equivocation. A Catholic equivocater could lie and tell the Protestants what they wanted to hear, but God would know that what the Catholic said was really the truth in another sense. Later in the play, the Witches, serving the devil, equivocate with Macbeth. For example, they tell him that he has no need to fear until Birnham wood comes to his castle. It sounds like they mean that he will never have a reason to fear, because trees can't walk, but it turns out that men can carry branches they have cut, so that the "wood" comes to the castle in that sense.
Later in the same scene, the Porter jokes with Macduff that liquor is an equivocator because it makes a man horny, but keeps him from doing anything about it. In the Porter's words, drink "provokes the desire, but it takes away the performance: therefore, much drink may be said to be an equivocator with lechery" (2.3.30-32). [Scene Summary]
When Macbeth goes to the witches to learn his fate, they call up apparitions, and the apparitions equivocate.
The first apparition is an "armed Head" (4.1.67, s.d.) which warns Macbeth to beware of Macduff. The surface meaning of the armed head is obvious: Macduff, in armor, will come at the head of an army to fight against Macbeth. The second meaning of the armed head becomes clear only at the end of the play, when Macduff, after defeating Macbeth in hand-to-hand combat, cuts off Macbeth's head and displays it to his soldiers.
The second apparition is a "bloody Child" (4.1.76, s.d.), which advises Macbeth to "Be bloody, bold, and resolute; laugh to scorn / The power of man, for none of woman born / Shall harm Macbeth" (4.1.81). This sounds like it means that no man can harm Macbeth, because every man is born of woman. Except Macduff. At the end of the play, in his last battle, Macbeth learns that "Macduff was from his mother's womb / Untimely ripp'd" (5.8.15-16). Cesarean section doesn't count. "Ripp'd" isn't "born."
The third apparition is a "Child crowned, with a tree in his hand" (4.1.86, s.d.). This apparition assures Macbeth that "Macbeth shall never vanquish'd be until / Great Birnam wood to high Dunsinane hill / Shall come against him" (4.1.92-94). To Macbeth it's as though he's just learned that he will never be defeated, but the appearance of the apparition tells a different story. Macbeth himself recognizes that the crowned child is the son of a king, and it's generally agreed that the child stands for Malcolm, who is the son of Duncan. Of course, by the end of the play, Macbeth is dead, and Malcolm is about to be crowned king. Also, the tree carried in the child's hand shows just how Birnam wood will come to Macbeth's castle -- it will be carried by soldiers using branches and saplings as camouflage. [Scene Summary]
Having gotten the idea that his father might be considered a traitor, the Son of Macduff asks his mother what a traitor is, and she replies, "Why, one that swears and lies" (4.2.47). "One that swears" is not a person who curses, but someone who swears to tell the truth or swears allegiance to his king. A person who "swears and lies" is one who swears an oath without meaning to keep the oath. This passage is often taken as an allusion to Henry Garnet's doctrine of equivocation, but Macduff's son makes a joke which says that almost everyone "swears and lies" at some time. [Scene Summary]
When Macduff is in England to get Malcolm's support for a war against Macbeth, Ross enters the scene and tells of all the terrible things that Macbeth has done in Scotland. Macduff asks about his wife and children, and Ross says that they are "well." Macduff repeats the question, asking "The tyrant has not batter'd at their peace?" (4.3.178). Ross answers with an equivocation: "No; they were well at peace when I did leave 'em" (4.3.179). We know that Macbeth has not "batter'd at their peace," he's killed them, and that they are "well at peace" because they are gone from this world to the peace of heaven. Ross has good intentions; he wants to spare Macduff's feelings, at least for the moment. [Scene Summary]
As the forces under Malcolm's command approach Dunsinane, a messenger tells Macbeth he has seen Birnam woods move. Macbeth begins to be afraid, and says that he suspects "the equivocation of the fiend / That lies like truth. 'Fear not, till Birnam wood / Do come to Dunsinane,' and now a wood / Comes toward Dunsinane" (5.5.42-45). [Scene Summary]
In his last battle Macbeth boasts to Macduff that he cannot be harmed by man "of woman born." Macduff then tells him that he was "from his mother's womb / Untimely ripp'd" (5.8.15-16). At this, Macbeth curses Macduff, then adds, "And be these juggling fiends no more believed, / That palter with us in a double sense; / That keep the word of promise to our ear, / And break it to our hope" (5.8.19-22). To "palter" is to equivocate. Thus Macbeth promises -- too late -- to never again believe in devilish equivocation.