What are the peripeteia and catharsis in the play Macbeth?

"Perepetia" is when a Greek hero realizes everything he believed in is wrong. "Catharsis" is  defined as

1. A purifying or figurative cleansing of the emotions, especially pity and fear, described by Aristotle as an effect of tragic drama on its audience.
2. A release of emotional tension, as after an overwhelming experience, that restores or refreshes the spirit.

In Macbeth there are two key points where he experiences Perepetia. The first occurs just after the killing of Duncan and before Macbeth arranges the murderers to kill Fleance and Banquo. Macbeth says he is "I am in blood / Stepp'd in so far, that, should I wade no more, / Returning were as tedious as go o'er" (3.4). Macbeth was undecided about killing Duncan at first, remember, and it was his wife who pressured him into doing the deed.

The second instance of perepetia is MacBeth`s infamous

To-morrow, and to-morrow, and to-morrow,
Creeps in this petty pace from day to day,
To the last syllable of recorded time;
And all our yesterdays have lighted fools
The way to dusty death. Out, out, brief candle!
Life's but a walking shadow, a poor player
That struts and frets his hour upon the stage
And then is heard no more. It is a tale
Told by an idiot, full of sound and fury
Signifying nothing." (5.5.17-28)

Here Macbeth is at his absolute lowest, realizing that his play is over and his actions have amounted to little more than the rantings and strutting of himself, the poor player.

Catharsis is experienced at the end of the play when the audience sees justice done and balance restored when Macduff beheads the traitorous King and mounts his head upon a pike for all to see.

What is the role of catharsis in Macbeth?

Another way in which many of the audience of Shakespeare's Macbeth pity the tragic figure of Macbeth in his destructive obsession with ambition.  When he is made Thane of Cawdor by King Duncan in Act I, he is a hero, admired for his bravery and skill.  But, like so many, he becomes seduced by ambition and the preternatural world as well, losing his integrity, his wife, and, finally, his life.  The temptress of ambition takes from him the essence of his being as a human, leaving him with a meaningless life.  Sadly, Macbeth realizes this in his soliloquy in Act V as he speaks of life as

...a poor player 
That struts and frets his hour upon the stage
And then is heard no more.  It is a tale
Told by an idiot, full of sound and fury,
Signifying nothing (5.5.24-28)

This condition of the seductiveness of power is also one that is feared by people, for so often the attainment of power leaves people empty as they have sold their souls to the other world of evil, sacrificing much of their humanity as does Lady Macbeth in her life as "a walking shadow." 

In Act IV, Scene 3, it is this very fear of the evil of power and its effects that Malcolm exhibits in his distrust of the motives of Macduff who has come to England to seek Malcolm's help in restoring the rightful ruler to Scotland.  For, he wonders if Macduff is a paid agent of Macbeth; he questions Macduff as a result and feigns corruption in himself in order to elicit Macduff's true feelings.  When Macduff exclaims, "O, Scotland!" Malcolm realizes that the man loves his country and is loyal, so he tell Macduff the truth about himself.

As you know, katharsis is the purgation of pity and fear.  It is what the audience should feel after a tragedy.  Macbeth is wholly tragic, and it presents a tragic hero who is so ambitious that he would pass as heroic today.  He might be a head of state or a corporation.

So, at the end of Macbeth, we pity Macbeth because he showed incredible courage to achieve his goals and dreams.  He refused to settle for his station according to the Great Chain of Being.  Instead, he defied fate and the gods to become King.  Granted, he murdered to do it, but as Machiavelli said, "The ends justified the means."  So, on a purely political level, Macbeth was very ingenious.

We also fear that Macbeth's fate may be our own.  We fear that we may die alone defending an empty castle, so to speak.  We fear that we and our spouse may become disillusioned, if not sleepless and crazy from the nothingness of life.  As Macbeth's soliloquy states, we may dread "tomorrow" and the "petty pace of day to day" without reaching our dreams.  Or, having reached them, regret them as folly.