Elizabeth and Zechariah are the parents of John the Baptist. Much like Abraham and Sarah (remember from Genesis about a month ago?) they were super old and were unable to have children. Zechariah was a faithful priest of God and he was chosen by lot (remember from Saul being selected as king?) to go in and light the incense in the sanctuary of the Temple. Once in the Temple an angel came and told him that Elizabeth and him were going to give birth to a baby who would go before the Messiah.
Being a priest Zechariah would know the Torah (the book of the law) which would include the life of Abraham and would know how he and Sarah gave birth in their old age. But Zechariah had a hard time believing that this would happen to him and Elizabeth. He expressed this doubt to the angel who did not like his lack of faith. He therefore made Zechariah unable to talk until the birth of his son.
In contrast to Zechariah's response we see Mary's response. She too questions how she can become pregnant when she is a virgin but the angel doesn't get mad at her. Zechariah had extensive knowledge of Scripture and God had worked in such a way before. Mary wouldn't have the same knowledge of Scripture and God hadn't worked in that way before. Which would be easier for you to believe? An 80 year old having a baby or a virgin having a baby?
Mary's acceptance of God's will shows her servant heart. She had a lot to lose being pregnant our of marriage. Her reputation would be ruined, her pending marriage to Joseph could be called off, and she be shamed for the rest of her life but she accepted the message given to her and trusted in God. She ran off to Elizabeth to see how she was and at the sound of Mary's voice, John in Elizabeth's womb leaped with joy. Little John was filled with the Holy Spirit even in the womb and celebrated the saviour of the world being near him.
Mary sees that everything the angel told her was true and despite how her world would perceive her Mary praises God. She doesn't look at what people will think of her but has an eternal perspective. She praises that "he took notice of his lowly servant girl, and from now on all generations will call me blessed" (Luke 1:48). She knew that through her God was going to change the world.