I'll try to add sources as time permits.
This book is not mentioned explicitly anywhere else in the Bible, and there are a number of theories found in the commentaries.
Rashi, Rashbam, Leclerc, and Luzzatto (and presumably many others, all to Numbers 21:14) support the theory that this was not an actual book that was written, but rather it simply refers to this fragment of the Bible narrative.
Among more modern authorities, N. H. Tur Sinai claims here that it never existed.
Ibn Ezra to Exodus 17:14 claims that this could have been the book that Moses wrote for Joshua at the battle with Amalek. However, in Numbers 21:14, he suggests that it may have been written in the times of Abraham. Similar claims are found in various other commentaries.
(Wikipedia quotes David Rosenberg in dating the book to ~1100BC, and Joseph Barber Lightfoot that it is another name for the Biblical Book of Jasher. However, no sources are provided for either claim.)
NZY Berlin suggests that it was a book chronicling the battles fought and happening as the Israelites traveled towards Canaan.
Nahmanides, Abarbanel and Mendelssohn (see Luzatto for a rebuttal) seem to understand that these were books written by other nations at that time about known wars, and that it was not Jewish in nature.
Some understand that it contains mainly events that took place, whereas others seem to understand that it was a book of songs and poetry.
Side note: In this CBQ article by Duane Christensen, he attempts to reconstruct the text that we have in order to make any sense out of it, but notes that some have dismissed it as "beyond reconstruction".