The first account is the judgment account, in this account having a king a judgment brought down upon the people as a curse for their rejection of God.
1 Samuel 8:4-22
Then all the elders of Israel gathered themselves together, and came to Samuel unto Ramah, and said unto him, "Behold, thou art old, and thy sons walk not in thy ways: now make us a king to judge us like all the nations." But the thing displeased Samuel, when they said, "Give us a king to judge us."
And Samuel prayed unto the LORD. And the LORD said unto Samuel, "Hearken unto the voice of the people in all that they say unto thee: for they have not rejected thee, but they have rejected me, that I should not reign over them. According to all the works which they have done since the day that I brought them up out of Egypt even unto this day, wherewith they have forsaken me, and served other gods, so do they also unto thee. Now therefore hearken unto their voice: howbeit yet protest solemnly unto them, and shew them the manner of the king that shall reign over them."
And Samuel told all the words of the LORD unto the people that asked of him a king. And he said, "This will be the manner of the king that shall reign over you: He will take your sons, and appoint them for himself, for his chariots, and to be his horsemen; and some shall run before his chariots. And he will appoint him captains over thousands, and captains over fifties; and will set them to ear his ground, and to reap his harvest, and to make his instruments of war, and instruments of his chariots. And he will take your daughters to be confectionaries, and to be cooks, and to be bakers. And he will take your fields, and your vineyards, and your oliveyards, even the best of them, and give them to his servants. And he will take the tenth of your seed, and of your vineyards, and give to his officers, and to his servants. And he will take your menservants, and your maidservants, and your goodliest young men, and your asses, and put them to his work. He will take the tenth of your sheep: and ye shall be his servants. And ye shall cry out in that day because of your king which ye shall have chosen you; and the LORD will not hear you in that day."
Nevertheless the people refused to obey the voice of Samuel; and they said, "Nay; but we will have a king over us; that we also may be like all the nations; and that our king may judge us, and go out before us, and fight our battles."
And Samuel heard all the words of the people, and he rehearsed them in the ears of the LORD.
And the LORD said to Samuel, "Hearken unto their voice, and make them a king."
And Samuel said unto the men of Israel, "Go ye every man unto his city."
The second account is what I would call the mercy account. In this account, God is giving the king as mercy upon Israel:
1 Samuel 9:15-16
Now the LORD had told Samuel in his ear a day before Saul came, saying, "To morrow about this time I will send thee a man out of the land of Benjamin, and thou shalt anoint him to be captain over my people Israel, that he may save my people out of the hand of the Philistines: for I have looked upon my people, because their cry is come unto me."
In this second account, the king appears to be a great blessing for the people of Israel, whereas in the first account, the king appears to be a curse. I guarantee you that if these two accounts were in two different books, we would be told that this was a Biblical contradiction. But the fact of the matter is that both aspects are true. The king was a blessing and a curse. The king brought judgment and mercy.
There is a third account as well, one that is almost neutral, although it tends toward the critical side:
1 Samuel 10:17-25
And Samuel called the people together unto the LORD to Mizpeh; and said unto the children of Israel, "Thus saith the LORD God of Israel, I brought up Israel out of Egypt, and delivered you out of the hand of the Egyptians, and out of the hand of all kingdoms, and of them that oppressed you: and ye have this day rejected your God, who himself saved you out of all your adversities and your tribulations; and ye have said unto him, Nay, but set a king over us."And if this third account were in a third book, we'd probably be told that this third account was a later account that smoothed some of the harshness of the first account. But, of course, it's all one book.
Now therefore present yourselves before the LORD by your tribes, and by your thousands. And when Samuel had caused all the tribes of Israel to come near, the tribe of Benjamin was taken. When he had caused the tribe of Benjamin to come near by their families, the family of Matri was taken, and Saul the son of Kish was taken: and when they sought him, he could not be found.
Therefore they enquired of the LORD further, if the man should yet come thither. And the LORD answered, "Behold, he hath hid himself among the stuff."
And they ran and fetched him thence: and when he stood among the people, he was higher than any of the people from his shoulders and upward.
And Samuel said to all the people, "See ye him whom the LORD hath chosen, that there is none like him among all the people?"
And all the people shouted, and said, "God save the king."
Then Samuel told the people the manner of the kingdom, and wrote it in a book, and laid it up before the LORD. And Samuel sent all the people away, every man to his house.
There is actually a reference to this event in another book. That book is Deuteronomy:
When thou art come unto the land which the LORD thy God giveth thee, and shalt possess it, and shalt dwell therein, and shalt say, "I will set a king over me, like as all the nations that are about me;" thou shalt in any wise set him king over thee, whom the LORD thy God shall choose: one from among thy brethren shalt thou set king over thee: thou mayest not set a stranger over thee, which is not thy brother.Isn't it remarkable that this book, written long before Samuel (notwithstanding the notions of the higher critics who deny Mosaic authorship) predicted that the people would demand a king and set regulations for that king, even without approving their rejection of God! This is a great example of one of the prophecies fulfilled in Scripture, even before the New Testament era.
But he shall not multiply horses to himself, nor cause the people to return to Egypt, to the end that he should multiply horses: forasmuch as the LORD hath said unto you, "Ye shall henceforth return no more that way."
Neither shall he multiply wives to himself, that his heart turn not away: neither shall he greatly multiply to himself silver and gold. And it shall be, when he sitteth upon the throne of his kingdom, that he shall write him a copy of this law in a book out of that which is before the priests the Levites: and it shall be with him, and he shall read therein all the days of his life: that he may learn to fear the LORD his God, to keep all the words of this law and these statutes, to do them: that his heart be not lifted up above his brethren, and that he turn not aside from the commandment, to the right hand, or to the left: to the end that he may prolong his days in his kingdom, he, and his children, in the midst of Israel.
And how was that King to be educated? He was assigned the prefect of the congregation of the doctrine of the faith and two bishops. Haha! No, that's not it. He was given the Torah - the Scriptures. He was supposed read from them, and learn to fear God from them. That's because the Scriptures are powerful and instructive. Simply by reading them, one can learn the fear of the Lord. They are true and convey the Truth to the reader.
God inspired that great prophet Samuel (who had a house, and sons - sorry ascetics) to write at least the first ten chapters of the book of 1 Samuel (or so it appears from 1 Samuel 10:25). And regardless of who wrote the powerful words of 1 Samuel, it should be clear that there is no contradiction amongst the narratives. The Israelites rejected God and got what they asked for - a king who was a burden to them. Nevertheless, at the same time the king was a savior to them, to save them out of the hand of the Philistines. And both the salvation and the judgment were intended by God in the one act of giving the people the king.
So when we read "conflicting" accounts in the gospels, we should be careful to realize that even seemingly conflicting accounts can sometimes be reconciled. Consequently, we should not be eager to find contradiction, but eager to find the harmony.
P.S. I wouldn't be the least surprised to discover that some higher critic has already tried to suggest that chapter 9 is from a different author than chapters 8 and 10. Of course, criticism of 1 Samuel tends to take a back burner compared with criticism of the Torah itself, but surely some skeptical scholar has taken aim at some of the other books.