Transverse wooden flute fashioned on the "simple-system", i.e. with six or so note-holes and few or no keys. The Irish flute is prized for its smooth and mellow tone.
The "Irish" flute was not designed with Irish traditional music in mind: it was the standard wooden concert flute of the early 1800s. In the 1850s the new Boehm-style flute came along. This had multiple keys, and was usually made of metal, frequently silver. It was a more reliable instrument for producing chromatic (12-note) scales as serious composition now required; it was louder and clearer too. Boehm-style flutes became the standard instruments for concert flautists.
Wooden flutes ("band" or "student" or "German", as they were called) remained popular with amateur musicians – particularly amongst the rising bourgeoisie. Mass production made them cheap (compared to most other respectable instruments) and more or less reliable. They retained something of their romantic associations (shepherds who blew half-naked through reeds), but an increasing "deeper" repertoire made them suitable instruments for "gentlemen". It is this convention that Brother Polycarp in the novel observes ("music in the tradition of Kuhlau and Briccialdi and like gentlemen of the transverse mode" Ch3.01).
During the same period the wooden flute gained a foothold in the Irish folk tradition. The story is told that a sudden abundance of discarded pre-Boehm flutes in pawn shops of the 1850s first attracted folk musicians to them. Irish music doesn't really require the full range of chromatic notes (cross-fingering will supply what half-steps are called for); and, as most traditional music was played in the home, loudness too was not essential. Soon the characteristic Irish flute style had evolved:
In the novel, the priest, Father Taylor, attempts to transform the genteel flute ensemble of Brother Polycarp into a "flute and drum" band.
The "flute and drum" marching band is common in Ireland to this day, and nearly always associated with politics – either loyalist or republican. But the instrument played is not a flute but a fife. The wooden flute cannot reliably play the third octave (the highest octave played on musical instruments). The fife, a shortened flute specifically designed to play high notes loudly, had been valued by armies for generations: its "shrill" note can be heard clearly above the deeper drums. It's not clear from the novel when the boys switched from a "flute band" to a "marching band" if they also switched instruments: if they did not, it's unlikely their flutes would have been heard at all.