The poem "Cat" was intended as being a poem in the marginalia in the Red Book, a piece written by Sam Gamgee as a touched-up version of an older piece of "comic bestiary lore of which Hobbits appear to have been fond" (The Adventures of Tom Bombadil, p. 232). It is a fairly short poem so I have quoted it below in it's entirety instead of writing a summary of it which would be only one sentence short.
Fastitocalon is based on an earlier poem. This version is a reduced text, something that would suit hobbits as part of old bestiaries, framed as an adaption of possibly more learned elvish lore. Tolkien was inspired by a fragment of Anglo-Saxon bestiary.
The poem Oliphaunt is in the Lord of the Rings story recited by Sam Gamgee in The Two Towers (Book IV, Chapter 3), the volume where it was first published. Sam Gamgee explains "That's a rhyme we have in the Shire [...] we have our tales too, and news out of the South, you know." This bestiary poem is related to an earlier one imagined by Tolkien called Iumbo.
This poem depicts nightmarish creatures called Mewlips and the depressing route one takes to their dwelling and what happens when you reach it. There exsists a precursor to The Mewlips, it is called Knocking at the Door: Lines Induced by Sensations When Waiting for an Answer at the Door of an Exalted Academic Person. Tolkien published that one under a pseudonym in Oxford Magazine (1937).
The title character of this book is none other than Tom Bombadil, who is featured in this poem. It is supposed to reflect poetry written by/being told by Hobbits in Buckland, who would be fairly well aquaintanced with Tom Bombadil, although they might not understand him perfectly. The poem have a fairly early origin, being concieved long before Frodo and his company meet Tom.
The Adventures of Tom Bombadil and Other Verses from the Red Book contains 16 poems, each with their own plot. It was first published in 1962. I have read a 2014 edition of the book, edited by Christina Scull and Wayne G. Hammond. It contains extra content, such as earlier versions of the poems, and commentary.