Short answers, pulled from the story.
The Red Book of Westmarch is a fictional manuscript that Tolkien uses as a found-manuscript device to explain the origins of his legendarium. In the fiction, it is a collection of writings recounted by the characters themselves. Those writings recount the events of The Hobbit and The Lord of the Rings.
Bilbo Baggins began it as his own memoir before Frodo Baggins expanded it into a record of the War of the Ring. Frodo's gardener, Samwise Gamgee, finished the remaining work, and unnamed hands in Westmarch later added a fifth volume of hobbit genealogies.
The name comes from its red leather binding and casing. It also comes from its having been housed in Westmarch, a region of Middle-earth next to the Shire. In reality, Tolkien modeled the name on the real Red Book of Hergest, a 15th century Welsh manuscript collection identified by scholar Mark T. Hooker.
The Thain's Book was the first copy of the Red Book. King Elessar of Arnor and Gondor commissioned it, and Thain Peregrin the First carried it to Gondor. A later copy was written by the scribe Findegil and stored at Great Smials. It was the only version that contained the whole of Bilbo's Translations from the Elvish.
According to Tolkien's own framing, Bilbo's translation of Elven histories and legends within the Red Book corresponds to The Silmarillion. Frodo's journey corresponds to The Lord of the Rings, and Bilbo's journey corresponds to The Hobbit.
In Peter Jackson's The Fellowship of the Ring, the book There and Back Again supplies the basis for the voiceover in the "Concerning Hobbits" scene. In the film, the book Bilbo hands to Frodo is subtitled A Hobbit's Tale rather than A Hobbit's Holiday. The Red Book itself appears in full at the end of The Return of the King.