Explain which book of Paradise Lost is the most important between I, II, IX, or XII?
An answer as to which Book in Milton's epic poem is the most important is going to be a matter of opinion. All of them are equally important and it's hard to make a case for one being more valuable than another. However, since substantiated opinion seems to be the only way to go on this one, I think that Book I is the essential because it establishes so many elements of the work. The exposition of the first book opens in media res, in the middle of the action. Milton uses the first book in order to “justify the ways of God to men" and helps to establish the emotional disposition that end up framing the narrative. This becomes another reason why Book I can be seen as the most important one in the epic poem.
Book I is also important because of its introduction to Satan. Book I helps to establish that the battle between Satan and God has been ongoing for some time. Book I is also important because it helps to establish the characterization of Satan, himself. It becomes clear that Satan is the way he is because of his refusal to submit to the will of God. Satan's unwillingness to accept the rule of God is established in the first book, something that continues throughout the epic poem:
For who can yet believe, though after loss,
That all these puissant legions whose exile
Hath emptied Heav'n shall fail to re-ascend,
Self-raised, and repossess their native seat?
Book I is important because it reveals how Satan's sole pleasure exists in his desire to destroy. It is Milton's genius to show that malevolence and evil exist when there is no creative force to bind it. Satan is shown in Book I to simply focus on "eternal war" against the divine. The characterization offered in Book I suggests that Satan has nothing but disdain for God: "Better to reign in Hell than serve in Heaven." This characterization is established in Book I, from which all other action originates. The idea of Satan never wishing to yield to God helps to guide the entire epic poem and is first introduced in Book I, helping to make this book one of the most important, if not the most important, in Milton's work.