Banquo's son, Fleance, is able to escape Macbeth's clutches. Macbeth and Banquo were both given prophecies by the witches, and Macbeth's have come true. Given that, Macbeth worries that Banquo now suspects him in the murder of King Duncan. In addition, the witches prophecy for Banquo was "thou shalt get kings, though thou be none," ( I, iii) which means that while Banquo will never be king, his son, or sons, will. Banquo has a son, Fleance.
To rid himself of worry, Macbeth sends murderers to kill Banquo and Fleance. They partially succeed; they kill Banquo, but Fleance escapes. Fleance alive can cause Macbeth a great deal of trouble - after all, the prophecy for his sons becoming king one day still holds true for Fleance, who might himself have sons. Macbeth realizes this fact when he tells his wife, "We have scotch'd the snake, not kill'd it," (III, i), but there is nothing he can do at this point.