According to the author of the book of 1 Chronicles, this verse is telling us about a list of families and their names being written down during the time when Jotham was the king of Judah and Jeroboam was the king of Israel. It's like a family tree from a long time ago, showing who belonged to which family.
1 Chronicles 5:17: All these were reckoned by genealogies in the days of Jotham king of Judah, and in the days of Jeroboam king of Israel.
1 Chronicles 6:1 lists Levi as one of the tribes in genealogies, whereas 1 Chronicles 5:17 only pertains to Reuben, Gad, and the half-tribe of Manasseh.
1 Chronicles 6:1: The sons of Levi; Gershon, Kohath, and Merari. [Gershon: or, Gershom]