King Saul, from Wikimedia Commons
While Samuel was a good judge, his two sons were apparently a couple of crooks; caring mostly about money, taking bribes, and having bad judgment in general. So, the Israelites didn't want these two guys running the show. Instead, they wanted to have a king to judge them.
Samuel was a little butthurt about this and consulted old Yahweh. Yahweh said by wanting a king, they are rejecting Me, and this is made most abundantly clear by the fact that they are still worshiping other gods!
Yes, the Israelites are still worshiping other gods! It's almost like this one isn't obvious enough for them.
Yahweh instructs Samuel to let the people know what a king entails, and Samuel does as he is told.
Sammy boy let's the Israelites know that a king will take all their firstborn sons and use them to ride his chariots in battle, or as foot soldiers; and that a tenth of everything they own will belong to the king.
The Israelites didn't mind that all, and still wanted to have a king to rule over them. Samuel reports this to the Creator of the Universe, and so He instructs Samuel to appoint a king. The chapter ends with Samuel calling out to the men of the city, "Go ye every man unto his city."