Nine sealed monsters numbered one to nine by tail count — power scaling made visible enough that a child can read it off a silhouette.
Filed by its count, not its title
Reduced as a phrase, “The Nine Tailed Beasts” would sit on The Telling — but that would be reducing wording this reference chose. The set is filed under 9 · The Closing instead, because 9 is what its author chose, and a chosen number outranks an imposed one.
What it is
Nine enormous chakra constructs, each sealed inside a human host, distinguished by how many tails they have. The One-Tail through the Nine-Tails; strength rises with the count.
What the number does
This is the cleanest example in manga of a numbered set solving the medium's hardest problem. Shōnen constantly needs a reader to know, instantly, who is stronger. Dialogue is slow, power levels are inelegant, and a fight that has to explain itself has already failed. Counting tails makes the answer visible in a single panel, at any distance, in black and white.
The consequence is that the set is also a countdown. Once the ranking is legible, the reader knows exactly how much story is left in it — and a villain collecting the beasts in ascending order is a plot that narrates its own remaining length.
Nine, and what it borrows
The Nine-Tails is the Kyūbi no Kitsune of Japanese folklore, where a fox is said to grow a tail per century and reaches nine at the height of its power. Kishimoto did not invent the ranking, only the eight below it: the folklore supplied the top of the ladder, and the manga built downward to meet it.