Elizabeth was apparently older than Mary, and although married for a long time to the Levite priest Zechariah (Elizabeth was also of the tribe of Levi - see Levites), they remained childless until God answered their prayer. Just after Mary was told by an angel (Gabriel announced the coming births of both Jesus Christ and John the Baptist) that she would become the mother of Jesus, she visited Elizabeth who was by then only three months away from giving birth to John the Baptist i.e. John the Baptist was about six months older than Jesus (and both of them were executed at about the same time - Jesus crucified, John beheaded). The Bible records that Mary remained with Elizabeth for about three months, until about the time of John's birth, when Mary herself would then have been six months away from the birth of Jesus.
The Bible plainly states that both Elizabeth and Mary were filled with the Holy Spirit, at a time when Christians did not yet even officially exist - Jesus Christ had not yet been born. Elizabeth was a Levite, Mary was a Jew. The record of the start their visit is of much love and joy:
"In those days Mary arose and went with haste into the hill country, to a city of Judah [today identified with the town of Ein Karem, in photo], and she entered the house of Zechariah and greeted Elizabeth. And when Elizabeth heard the greeting of Mary, the babe leaped in her womb; and Elizabeth was filled with the Holy Spirit and she exclaimed with a loud cry, "Blessed are you among women, and blessed is the fruit of your womb! And why is this granted me, that the mother of my Lord should come to me? For behold, when the voice of your greeting came to my ears, the babe in my womb leaped for joy. And blessed is she who believed that there would be a fulfilment of what was spoken to her from the Lord." (Luke 1:39-45 RSV)
The account of Elizabeth's life is that of faith and piety. A very good example for us all.
Fact Finder: Who named John, later known as John the Baptist?
(a) his mother, Elizabeth (b) his father, Zechariah
Luke 1:59-60 (John almost became known as Zechariah the Baptist!)