In the days of Herod, King of Judea, there was a priest named Zechariah of the priestly division of Abijah; his wife was from the daughters of Aaron, and her name was Elizabeth.  
Both were righteous in the eyes of God, observing all the commandments and ordinances of the Lord blamelessly.
But they had no child, because Elizabeth was barren and both were advanced in years.
Once when he was serving as priest in his division's turn before God,
according to the practice of the priestly service, he was chosen by lot to enter the sanctuary of the Lord to burn incense.
Then, when the whole assembly of the people was praying outside at the hour of the incense offering,
the angel of the Lord appeared to him, standing at the right of the altar of incense.
Zechariah was troubled by what he saw, and fear came upon him.
But the angel said to him, "Do not be afraid, Zechariah, because your prayer has been heard. Your wife Elizabeth will bear you a son, and you shall name him John.
And you will have joy and gladness, and many will rejoice at his birth,
for he will be great in the sight of (the) Lord. He will drink neither wine nor strong drink. He will be filled with the holy Spirit even from his mother's womb,
and he will turn many of the children of Israel to the Lord their God.
He will go before him in the spirit and power of Elijah to turn the hearts of fathers toward children and the disobedient to the understanding of the righteous, to prepare a people fit for the Lord."
Then Zechariah said to the angel, "How shall I know this? For I am an old man, and my wife is advanced in years."
And the angel said to him in reply, "I am Gabriel, who stand before God. I was sent to speak to you and to announce to you this good news.
But now you will be speechless and unable to talk until the day these things take place, because you did not believe my words, which will be fulfilled at their proper time."
Meanwhile the people were waiting for Zechariah and were amazed that he stayed so long in the sanctuary.
But when he came out, he was unable to speak to them, and they realized that he had seen a vision in the sanctuary. He was gesturing to them but remained mute.
Then, when his days of ministry were completed, he went home.
After this time his wife Elizabeth conceived, and she went into seclusion for five months, saying,
So has the Lord done for me at a time when he has seen fit to take away my disgrace before others.

Gospel commentary

Saint Maximus of Turin (?-c.420), Bishop
Sermon 5 ; PL 57, 863

It was prayer, not sexual desire, that brought about John's conception. Elizabeth's womb was already past the age of childbearing, her body had lost all hope of conceiving, yet, in spite of this state of hopelessness, the prayer of Zachariah permitted her aged body to bear fruit once more. It was grace, not nature, that conceived John. He could not but be a saint, this child whose birth resulted less from an embrace than from prayer.

However, we shouldn't be surprised that John was worthy of so glorious a birth. The birth of Christ's forerunner, of him who made straight his way, had to show some resemblance to that of the Lord our Savior. Thus, if our Lord was born of a virgin, John was conceived by an old and barren woman (…) And we do not wonder any the less at Elizabeth for conceiving in her old age than at Mary for giving birth in virginity.

To my mind there is something symbolic in this: John represented the Old Testament; he was born of the blood, already cold, of an aged woman. Whereas Our Lord, who announced the Good News of the Kingdom of heaven, is offspring of vigorous youth. Mary, aware of her virginity, wonders at the child curled up in her womb; Elizabeth, aware of her age, blushes at her womb swollen by pregnancy; as the evangelist says: "She went into seclusion for five months." We are to admire, too, the fact that the same archangel, Gabriel, announced both births: he brought consolation to Zachariah, who remained incredulous; he came to Mary, whom he found believing (Lk 1:26f.). The first lost his voice for having doubted; the second, because she believed, at once conceived the Word, the Savior.