In the days of Herod - See the notes at Matthew 2:1.
Of the course of Abia - When the priests became so numerous that they could not at once minister at the altar, David divided them into 24 classes or courses, each one of which officiated for a week, 1 Chronicles 24. The class or course (shift) of Abia was the “eighth” in order, 1 Chronicles 24:10. Compare 2 Chronicles 8:14. The word “course” means the same as “class,” or order. The Greek-based word “Abia” is the same as the Hebrew-based word “Abijah.”
His wife was of the daughters of Aaron - A descendant of Aaron, the first high priest of the Jews; so that “John the Baptist” was descended, on the father’s and the mother’s side, from priests. Our Saviour was not on either side. John would have been legally entitled to a place among the priests; our Saviour, being of the tribe of Judah, would not.