We are told that Odysseus got his bow from someone he met in Lacedaemon. The man was named Iphitus, son of Eurytus. Odysseus traded a sword and a spear to Iphitus in return for the bow.
We are never really told very much about the physical appearance of the bow. Odysseus looks to see if worms have eaten its horns. And we know it has a centerpiece, where he rests his arrow, but we are not told anything more about it. The only thing the poet focuses on with the bow is how hard it is to string it, how only Odysseus is able to manage that feat.