Mary, pregnant with Jesus and Elizabeth, pregnant with John the Baptist meet in Hebron, south of Jerusalem, immediately after the Annunciation. The cousins met so that Christ, still a fetus at the time, would bring divine grace to both Elizabeth and her unborn child. Sensing Christ's divine presence, John leapt for joy as he was cleansed from original sin and filled with divine grace. (Luke 1:44) This leads Elizabeth to also recognize His presence and responds in a loud voice praising God:
"Blessed art thou among women, and blessed is the fruit of thy womb." (Luke 1:42)
It is from this speech that we get one of the Church's most iconic prayers, the Hail Mary.
At this point, Mary responds with another speech which has become a very important Christian prayer which is used in the Liturgy of the Hours' Evening Prayers, the Magnificat:
My soul magnifies the greatness of the Lord,
my spirit rejoices in God, my Savior.
He looks on His servant in her lowliness;
henceforth all ages will call me blessed.
The Almighty works marvels for me.
Holy His name!
His mercy is from age to age,
on those who fear Him.
He puts forth His arm in strength
and scatters the proud-hearted.
He casts the mighty from their thrones
and raises the lowly.
He fills the starving with good things,
sends the rich away empty.
He protects Israel, His servant,
remembering His mercy,
the mercy promised to our fathers,
to Abraham and his sons for ever. (Luke 1:46–55)
It was at this meeting that Mary exercised her function as Mediatrix between God and man for the first time
This feast is of medieval origin and started by the Franciscans even before St. Bonaventure recommended it and the Franciscan chapter adopted it in A.D. 1263. In 1389, Pope Urban VI, hoping to end the Great Western Schism, inserted it in the Roman Calendar thus giving it to the Universal Church.