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).
This is a nonsensical poem composed by Samwise Gamgee and it is also found in The Lord of the Rings. Earlier versions by Tolkien called the poem Pero & Podex or Root of the Boot and had been written/published between 1920-1936.
This poem is written by Samwise Gamgee and he has possibly based some of the characters on real hobbits. It features a lonely Troll befriending Perry-the-Winkle, a young hobbit. In ~1928 Tolkien wrote an earlier version to this poem called The Bumpus for a series of works.
This short poem is described as having been written into the marigins of the Red Book. It is supposed to depict a hobbit's idea of what an elf-maiden is. A precursor of the poem, called The Princess Ni, was published in the collection Leeds University Verse 1914-24.
Errantry is a poem made by Bilbo, possibly on early days after returning from his journey. The poem is a bit nonsensical and follows a mariner. An earlier version of this poem, with the same name, was published in the Oxford Magazine the 9th November 1933. A later version of the poem is Eärendil Was a Mariner which is also by Bilbo, in The Lord of the Rings.
For the second poem in this publication, Tom is yet again the main character. This time the reader follow him on a boating trip along the Withywindle and his journey to meet Farmer Maggot. In this story we encounter familar places in Buckland and the Shire. The idea is that the events set out occured some time after Frodo and the other Hobbits visited Farmer Maggot and Tom Bombadil and that the poem is composed by Buckland hobbits.
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.