In Eastern tradition, it is often called The Meeting of Our Lord and God and Savior, Jesus Christ, because the hymns emphasize the incarnate God, the Lord Jesus Christ, meeting with his people, in the persons of the Prophet Simeon and the Prophetess Anna.
In the Gospel of Luke 2:22-35, Mary and Joseph took the infant Christ to the Temple in Jerusalem. He was received in the arms of the elder Simeon, who then prayed, "Now let Thy servant depart in peace... for I have seen Thy salvation." This was one of the things that Mary "pondered in her heart"—the fact that others recognized that her Son was the Messiah. More on The Meeting of the Lord in the Temple
Please note that the content of this post primarily consists of articles available from Wikipedia or other free sources online.