In Shakespeare's Macbeth, Macbeth has murdered Duncan so that he can become king. He has been won over by the witches' predictions and urged forward by his wife's malicious nagging and insults. Macbeth's tragic flaw is his ambition, which he cannot (or will not) control.
In Act One, while returning from the battlefield, Macbeth and his good friend Banquo meet three witches. The "weird sisters" make predictions for each man. However, where Banquo brushes them off, Macbeth takes their words to heart—for one prediction promises that he will be king one day. Rather than waiting for fate to run its course, Macbeth decides to move things along and kill Duncan. It does not mean that Macbeth will be next in line to the throne, but we can anticipate that if he is willing to kill Duncan, he will have no difficulty in killing Duncan's sons.
Once Duncan is dead and his sons have fled in fear of losing their own lives, Macbeth becomes King of Scotland. However, two things rankle Macbeth. First, Banquo was present when the witches made their first set of predictions. Macbeth suspects (and rightly so) that Banquo will grow suspicious of the hand Macbeth may have had in bringing about the fulfillment of the predictions, specifically as it pertains to Duncan's murder. Macbeth also is aware that the witches predicted that Banquo would father a line of kings, but that Macbeth will have no sons to pass the throne on to. So Macbeth sees that he has lost his soul to eternal damnation (for it is a mortal sin to kill a king) only to see Banquo's sons fall heir to that same throne.
Macbeth is afraid of Banquo. To be king is all well and good, but it means nothing if he cannot safely enjoy it. All he has done is for nothing if Banquo exposes Macbeth. Macbeth knows the character of this man who was once his closest friend: he knows that Banquo will do the right thing and nothing will sway him from his purpose. Banquo is not a man who would lie for friendship or personal profit. His "royalty of nature" dictates that he will do whatever is necessary to remain an honorable man. In that he cannot be moved, Macbeth is solely afraid of Banquo's ethical and moral nature.
To be thus is nothing,
But to be safely thus. Our fears in Banquo
Stick deep, and in his royalty of nature
Reigns that which would be fear'd. ’Tis much he dares, (55)
And, to that dauntless temper of his mind,
He hath a wisdom that doth guide his valor
To act in safety. There is none but he
Whose being I do fear… (III.i.52-59)
Macbeth decides to have Banquo killed. He is aware that Banquo and his son Fleance will be out riding and will not return until after dark, so Macbeth arranges for assassins to ambush Banquo and his son.
Macbeth is obsessed that he has given all that matters to a man (his soul) so that Banquo's descendants will one day rule Scotland. He is aware that he murdered "gracious Duncan" to create a scenario where Macbeth loses everything to someone else's benefit, and he no longer has any peace of mind.
For Banquo's issue have I filed my mind,
For them the gracious Duncan have I murdered,
Put rancors in the vessel of my peace
Only for them, and mine eternal jewel
Given to the common enemy of man,
To make them kings, the seed of Banquo kings!
Rather than so, come, Fate, into the list,
And champion me to the utterance! (69-76)
This is not the man Macbeth was at the start of the play. However, even as his murder of Duncan has taken its toll on his conscience, his ambition drives him ever forward—and now he plans to kill Banquo.
At the close of the segment above, Macbeth chooses to kill Banquo and his son; he will let fate decide what happens on a new battleground ("list")—which we can assume will most likely not end here. Macbeth not only fears that Banquo will expose him as a murderer, but that Macbeth has lost his soul to the devil ("the common enemy of man") only to guarantee that Banquo's "seed" will sit on the throne. In killing Banquo and Fleance, Macbeth hopes to do away with these threats.