Earlier in Act Three, Scene 3, Macbeth's assassins murder Banquo but allow Fleance to escape, making it possible for Banquo's descendants to one day inherit Macbeth's throne. In Act Four, Scene 1, Macbeth visits the Three Witches, and they conjure several enigmatic apparitions that portray Macbeth's future. After Macbeth sees the three apparitions, he demands to know if Banquo's sons will ever reign over the kingdom. The witches then show Macbeth a procession of eight kings. The last king in the procession is holding a mirror, and Banquo's ghost follows behind the line of kings. Macbeth is tormented to discover that the kings are all descendants of Banquo, including the seemingly endless line of kings he sees in the mirror. As Macbeth is looking into the mirror, Banquo's ghost smiles at him as he points to his royal descendants. Macbeth comes to terms with the weight of the prophecy and vows to act immediately upon his impulses. He descends deeper into tyranny and instantly issues an attack on Macduff's innocent family. Essentially, Banquo's revenge is that his descendants will become kings, which drastically unnerves Macbeth to the point that he begins to make irrational, tyrannical decisions. 

Concerning Shakepeare's Macbeth, I assume you're referring to the vision Macbeth sees that presents Banquo's heirs as kings. The vision presents the line of Stuart kings which culminates in James I, who was reigning at the time the play was first produced. 

Macbeth, of course, tries to change fate in Act 3 when he plans and orders the killing of Banquo and Fleance.  He succeeds in eliminating Banquo, but fails to eliminate Fleance.  This makes the ruling by Banquo's heirs possible.

The last figure shown in the vision is Banquo, holding a magic mirror that shows Macbeth many more kings.  Macbeth yells, "I'll see no more."  But the kings just keep coming.  And the blood-matted Banquo smiles at Macbeth while pointing to the kings. 

Macbeth asks the witches if this is so. 

These are the only specific reactions to the vision revealed.  But, in reaction to all of the visions shown him, and in reaction to the news that Macduff has fled to England, Macbeth determines to act immediately on his first thoughts:

The very firstlings of my heart shall be

The firstlings of my hand.

He will not think or hesitate before acting from now on.  And he keeps his word, immediately ordering the slaughter of Macduff's family.