“Athair” is the Irish word for “father”, and the Anglicized title “Fr. Jack Walsh” is often used as the title for this tune, as is “Tatter Jack Walsh”.
The tune is played in the D mixolydian mode and thus contains the same notes as a G Major scale. A cursory glance at the chords used in the sheet music should alert you to this. The two main chords used are D and C and this combination in a “D” tune invariably tells you it’s mixolydian.
The key signature is one sharp (F#). If I used two sharps (F# and C#) then there would be natural signs used for the C notes occurring in the tune.
Some tune books incorrectly notate the tune this way. Tony’s old copy of that excellent small collection “Allan’s Irish Fiddler” has it in D Major with natural signs for the C. Although the transcription still ends up with the same notes it is not very good “musical grammar”. There are two accidentals in the tune, both in the last bar of each section. In fact the last thee bars of each section are identical.