A little background on my question: As I understand it, the priesthood was not found among the Jews during the inter-testamental period (from the death of Malachi until the birth of John the Baptist) due to local apostasy. Obviously the Nephites still had the priesthood during this time which prevented a general apostasy – so if John the Baptist was born in a time and region where the priesthood didn’t exist, how and by whom was John ordained to the Aaronic Pristhood?
The Levitical (and Aaronic) Priesthood under the Law of Moses was based on lineage and worthiness, and it was still active during the intertestamental period. If we go to Luke 1:5, we see that John’s father, Zacharias was a priest (which he could only be if he were descended from Aaron – as indicated via the footnote for Abia) and that John’s mother, Elisabeth, was also descended from Aaron:
5 There was in the days of Herod, the king of Judæa, a certain priest named Zacharias, of the course of Abia: and his wife was of the daughters of Aaron, and her name was Elisabeth.
This means that John would have the right to the priest’s office as a descendant of Aaron. Although the Bible is silent on the details of what happened with John, it may have been similar to what was done with Aaron and his sons (and later other priests) who were washed, anointed with blood and oil, and consecrated priests (see Exodus 29). Whatever the case, we can assume from the message of the angel Gabriel before John was conceived (see Luke 1) and from the story of Christ’s baptism (see Matthew 3:13-17), that God the Father and Jesus Christ recognized John’s priesthood authority.
Also, we have this from the Bible Dictionary entry on John the Baptist:
Son of Zacharias and Elisabeth, being of priestly descent through both parents. This lineage was essential, since John was the embodiment of the law of Moses, designed to prepare the way for the Messiah and make ready a people to receive Him. He was the outstanding bearer of the Aaronic Priesthood in all history and was entrusted with its most noble mission.
I hope this clarifies for you the difference in holding the priesthood in John’s day versus our own.
For further reading about the Aaronic Priesthood, including how it changed after the Law of Moses was fulfilled, see the Bible Dictionary entries for Aaron and Aaronic Priesthood. And, for some in-depth reading about John the Baptist, you might enjoy “John the Baptist: The Miracle and the Mission” an article in the BYU publication Religious Educator.