I have a different interpretation of Macbeth's metaphor than the one offered by the first answerer, although I agree that it is about horses and horsemanship. In the first part of the soliloquy Macbeth thinks of all the reasons why he should not be thinking about murdering King Duncan. Then he decides that he has no justification for doing so, i.e., that he has no spur (reasonable excuse or even practical motive) to prick the sides of his intent (to go forward with the murder) except unreasonable, gross, dishonorable ambition which can ruin his reputation and lead to disaster. The part about o'erleaping itself suggests an overly reckless and impetuous rider leaping onto the back of his horse and falling off on the other side. His wife interrupts him before he can finish his last sentence. Instead of saying "And falls on the other side," the text has "And falls on the other--" with a dash signifying her interruption.

It is significant that he tells his wife:

We will proceed no further in this business.

He hath honored me of late, and I have bought

Golden opinions from all sorts of people,

Which would be worn now in their newest gloss,

Not cast aside so soon.

But Lady Macbeth uses all her powers of persuasion to overcome his doubts, fears, and scruples. The scene makes it obvious that Macbeth would not have committed the murder without the influence of his ambitious wife.

The quote "I have no spur to prick the sides of my intent, but only Vaulting ambition" comes from Macbeth's soliloquy in Act One, scene seven.  This is Macbeth's main speech in which he wonders if he should murder Duncan or not. 

In the final lines of the soliloquy, as he questions his motives, Macbeth says the "I have no spur" line; it his way of saying that his ambition motivates him to act.  The "spurs" are a metaphor for riding a horse.  If the rider wants the horse to move quickly, he kicks in his spurs, the sharp points attached to the heel of the boot.  In Macbeth's case, his ambition acts like the spurs do for the horse; his desire for power kicks in and motivates him to act, "vaulting" or leaping over obstacles in his way.