There are really two separate types of conflict in Shakespeare's Macbeth. The first type is external, in which characters struggle against other people or circumstances. The second type of conflict is internal, in which characters struggle primarily against their own natures. What makes the play so interesting is primarily the way Macbeth and Lady Macbeth struggle to overcome their better natures.
When Lady Macbeth observes the approach of Duncan, she says:
... Come, you spirits ... unsex me here,
And fill me from the crown to the toe topful
Of direst cruelty!
By this she means that as a woman she is naturally kindhearted, malleable, and nurturing but that to advance her plot and make Macbeth king, she needs to steel herself to act in an unwomanly manner by helping kill Duncan. Thus we see her struggle against her natural inclinations in this passage. This conflict though, eventually drives her mad.
Macbeth, who initially was described by Lady Macbeth as having a nature "too full o' th' milk of human kindness" becomes stronger and more evil as the play progresses, overcoming his internal conflicts, with each evil act making it easier to think and behave in an evil fashion.
The external conflicts consist of Duncan's justified suppression of the rebellion, Macbeth's murders of Duncan, Banquo, and Macduff's family, and the eventual overthrow of Macbeth by Malcolm and Macduff.