In Macbeth, the last spirit to rise from the cauldron in The Apparition Scene, is Banquo's ghost, holding a mirror. What does it signify?

It is important to remember that the prophecies of the
witches and the signs and visions that they summon up are deilberately meant to
obfuscate and present nebulous meanings that could be interepreted in number of ways.
Given this reality, we can conclude that there might be a number of different responses
to the meaning of this final symbol.

Let me start, however,
by one small correction. The stage directions of this play state that it is the final of
the eight kings that carries the mirror, rather than the ghost of Banquo, who appears
after these eight kings. Given the presence of Banquo at the end of this procession of
kings, perhaps we can interpret this final vision as refering to the way that the future
kings of Scotland will be descended from Banquo, even though he was not king himself.
Some critics suggested that the mirror was meant to reflect the image of King James I of
England, who claimed to be a descendant of Banquo. Either way, this final vision
reinforces the ironic truth of the witches prophecy of Banquo made in Act I scene 3,
when the Third Witch tells him "Thou shalt get kings, though thou be none." The vision
represents the inevitability of Macbeth's death without producing an heir, and the way
that, in spite of all he has done to seize the crown, he faces it being taken away from
him.