This article contains spoilers from the light novels that have not yet been revealed in the anime. Proceed at your own risk.
Hibike! Euphonium is a fictional book of etudes for euphonium composed by Shindo Masakazu. One of the few published copies belongs to Masakazu's daughter, Asuka Tanaka, which he sent to her along with his euphonium, when she was young. She constantly practices the pieces within the book when she has time. She highly cherishes it, as it is one of the only reminders of her father that she possesses asides from his silver euphonium.
One of the etudes in the book is also titled Hibike! Euphonium, and is Asuka's favorite. She is seen practicing it outdoors in the Episode 3 of the second season of the anime, and plays it again for Kumiko in Episode 9.
The melody of the Hibike! Euphonium is a recurring theme in the soundtracks of the anime's second season, and is particularly pronounced in track 28: 重なる心 (Overlapping Hearts), from the soundtrack of the first season.
Upon graduating, Asuka gives the book to Kumiko Oumae, saying that she now has more use for it than herself.
The book Hibike! Euphonium appears primarily in the third light novel, and in the second half of the anime's second season.