What is the significance of the show of eight kings given to Macbeth?
The significance of the parade of eight kings revealed to Macbeth in Shakespeare's Macbeth Act 4.1 is that it demonstrates that Banquo's heirs will rule Scotland, not Macbeth's. Ultimately, Macbeth will fail.
Once Fleance escapes his would-be murderers in Act 3, Macbeth cannot create the dynasty he is hoping to. Though he succeeds in having Banquo killed, Macbeth fails to have Banquo's heir killed. This is the beginning of the end for Macbeth, and possibly the climax of the play. The parade of kings highlights Macbeth's failure.
It also contributes to Macbeth's fear and anxiety, and to his further loss of control. And it highlights the oppositional elements in Macbeth's mind that conflict throughout the last three acts of the play: since his loss of control at the feast when he "sees" Banquo's ghost, Macbeth ping pongs back and forth between wanting to believe he is indestructible as the visions he sees seem to predict, and is knowing that the predictions are too good to be true and that he is doomed.
The parade of kings also, by the way, gives homage to the king that the play was produced for: James I, who was a descendant of Banquo.
The reason the apparition shows the eight kings is to bring to fruition the prophecy the witches gave to Banquo. Remember? Banquo will not be king, but his children will, and so will his children's children. This is the whole reason Macbeth wanted Banquo AND Fleance (his son) to be killed when he arranged the murderers to go after them.
The last king in the apparition holds a mirror to reflect a never-ending line of kings descended from Banquo. They all look like Banquo, too. By the way, this play was written for King James I of England who was supposedly descended from Banquo. This is why the character of Banquo is shown as such a good and honorable person, and why the never-ending line of kings is shown...King James I being one of them.
Macbeth's reaction? He is furious, of course. He does not want to be deposed. He is angry that Fleance escaped his murderers' grasp. It makes him uneasy to think that his own children will not ascend to the throne and that their will be no Macbeth Dynasty in Scotland.