What do the major occurrences of nature in Macbeth represent?
Since the murder of a king is an unnatural action, nature is affected. The report of the storm is evidence of nature's reaction to Macbeth's murder of Duncan. Order can only be restored by Malcolm taking his rightful place as king. It is only by the death of Macbeth that order will be restored. "Blood will have blood, they say."
From that point in the play until MacDuff kills Macbeth, the world of the play is "unnatural". Macbeth steeps himself in blood to keep covering up his crime. He kills his good friend, Banqueo since he knows that Banqueo suspects him. He tries to neutralize the prophesy that Banqueo will beget a line of kings, thus interfering with the supernatural. He kills the MacDuff family out of fear. Making war against women and children is unnatural.
Fear and suspicion become the way of life in Scotland. Friends are afraid of to speak their mind. MacDuff must convince Malcolm he is not a spy but sincere in his desire to help him.
All of this is unnatural. The references to nature in the play are all predatory for the most part. There are very few positive nature references. Ravens and violent storms are more the norm here.
It is interesting that the play deals with natural, unnatural and supernatural. Shakespeare ingeniously mixes them all together.
It is also necessary to realize that plots against the monarchs at this time were rampant. Numerous plots were hatched to oust Elizabeth I and the Gunpowder Plot which would have killed James I and a majority of the government and church leadership had just been discovered and thawarted. These issues were very real to Shakespeare's audiences. There is a reference to the Gunpowder Plot in the Porter's speech.
In the Elizabethean world, order was important and any disturbance of that order was considered unnatural.
Good answers so far. I'd add the unusual occurrences which happen preceding Duncan's death but we hear after the fact.
Rosse and the Old Man are conversing in Act II scene iv. The Old Man says he's lived seventy years and hasn't seen circumstances this strange in all his life. Ross agrees, remarking that though it is day, it is as dark as night (a solar eclipse, obviously).
ROSS: And Duncan's horses--a thing most strange and certain--
Beauteous and swift, the minions of their race,
Turn'd wild in nature, broke their stalls, flung out,
Contending 'gainst obedience, as they would make
War with mankind.
OLD MAN: 'Tis said they eat each other.
Clearly falcons should be attacking owls, not the other way around; and horses don't generally go crazy for no reason and then eat each other. Nature itself is obviously rebelling against the unnatural death of the king. (These occurrences are also found in other works, such as Julius Caesar, by the way.)
The idea of such odd disruptions in nature come from the idea that a king was ordained as such by God, and if anyone defiled that ordination (disrupted God's course of planned events) then His creation would cry out at the sin. These crazy things, plus the weather as discussed above, are indicators that God's man has been unnaturally dealt with--and the consequence for the murderer(s) is eternal damnation.
Major occurrences of nature in Macbeth help set the tone of scenes and also foreshadow events to come. For example, in Act 2 Scene 3, Macduff and Lenox arrive at Macbeth's castle, bringing news that there has been a terrible storm. Lenox says,
The night has been unruly: where we lay, our chimneys were blown down; and, as they say, lamentings heard i'th'air; strange screams of death, and, prophesying with accents terrible of dire combustion, and confus'd events, new hatch'd to th'woeful time, the obscure bird clamour'd the livelong night: some say, the earth was feverous, and did shake.
Immediately after this, they go to wake King Duncan and find him murdered. So, the information about the storm is a direct link to the murder of the King. The storm symbolically suggests that King Duncan's soul cried out the injustice inflicted upon it, unsettling the atmosphere of the place.