Macbeth's inner conflict should be obvious. He wants to become king but he doesn't want to kill King Duncan in order to succeed him. His wife understands his conflicting feelings. In Act 1, Scene 7, she taunts him:
Wouldst thou have that
Which thou esteem'st the ornament of life,
And live a coward in thine own esteem,
Letting "I dare not" wait upon "I would,"
Like the poor cat i' th' adage?
Banquo's conflict is more complicated. He is sure that Macbeth murdered Duncan in order to become king. Banquo doesn't know whether he should serve Macbeth and treat him as the lawful ruler, or whether he should turn against him like Macduff. Banquo knows his own life is in danger for at least two reasons. For one thing, he knows that Macbeth might be thinking of killing him before he can kill Macbeth. And furthermore, he must realize that Macbeth hates the thought of having murdered Duncan and sold his soul to the devil for the benefit of Banquo's descendants. Banquo could expedite fulfillment of the witches' prophesy by disposing of Macbeth the same way Macbeth disposed of Duncan. When the two men chance to meet in the dead of night in Act 2, Scene 1, they are both afraid of each other. Banquo says to his son:
Give me my sword!
At this point Banquo might feel tempted to go along with Macbeth, who begins sounding him about about joining in the assassination. With Banquo's help Macbeth could dispose of Duncan and his two sons, all on the same night. But Banquo decides to play wait-and-see. He tells Macbeth:
So I lose none
In seeking to augment it [his honor], but still keep
My bosom franchised and allegiance clear,
I shall be counseled.
Banquo may have toyed with the idea of killing Duncan, or killing Macbeth, himself. Killing Macbeth would not really be a crime or a sin, but only justice--since he could claim that Macbeth was an assassin and a usurper. Banquo must have also had the same fears as Macbeth about having someone kill him. He is apprehensive and suspicious; yet he has to maintain his composure in front of everybody. The witches' prophesy is somewhat favorable to him, but it seems as if he has to be dead before his progeny can become kings of Scotland.
I think that one of the most basic conflicts in Banquo lies in the drama's exposition. Banquo is conflicted with the witches' prophecies. On one level, he is conflicted because he does not accept the prophecies. Even though Banquo has been projected to give birth to a race of kings, he is not persuaded with the witches' prophecies: "oftentimes, to win us to our harm, / The instruments of darkness tell us truths, / Win us with honest trifles, to betray [us]." This line also indicates that Banquo is in conflict with how Macbeth is so taken in with the witches. Banquo is in conflict because he sees his friend so enamored with what the witches have said, while he is in denial about their veracity. This creates a fundamental conflict in Banquo. It represents a demarcation that Macbeth is willing to cross and one across which Banquo is unable. This becomes aspect of conflict within Banquo.
Macbeth's conflict lies in the fundamental moment of action and decision. In Act I, scene 7, Macbeth experiences conflict in having to kill Duncan. After being goaded by his wife and believing in the fated words of the witches, Macbeth experiences a conflict in terms of understanding the full implications of killing Duncan:
To our own lips. He's here in double trust:
First, as I am his kinsman and his subject,
Strong both against the deed; then, as his host,
Who should against his murderer shut the door,
Not bear the knife myself.
Macbeth experiences conflict with having to kill Duncan. He recognizes that it is against his very nature to do so. The conflict of having to "bear the knife myself" represents one of the first moments in which Macbeth is a caring human being. In the end, this conflict becomes significant because it shows equivocation. However, it bears significance because of how much he changes in the drama. This conflict becomes meaningful in light of the arc of Macbeth's characterization.