Hail Mary, full of grace, the Lord is with you.
Luke 1:28 "And coming to her, he said, "Hail, full of grace" (NAB: "highly favored one") [kecharitomene]! The Lord is with you."
Kecharitomene: favored by grace; suggests a permanent state of being "highly favored," thus "full of grace." God is infinite goodness. Mary is perfect created goodness, to the limit of her finite being.
Blessed art thou among women
Luke 1:41-42a "When Elizabeth heard Mary's greeting, the infant leaped in her womb, and Elizabeth, filled with the holy Spirit, cried out in a loud voice and said, "Most blessed are you among women..."
Luke 1:48 "For he has looked upon his handmaid's lowliness; behold, from now on will all ages call me blessed."
"Among all women": this Hebraicism suggests that Mary is the greatest of women. This must include Eve, whom God created immaculate, i.e. free of sin. Mary's preservation from sin by God's grace, both at her conception and throughout life, is what makes her greater than Eve, who fell from the grace in which God created her.
Blessed is the fruit of your womb, Jesus
Luke 1:42b "and blessed is the fruit of your womb."
Jesus is Mary's fruit. Good fruit does not come from anything but a good tree (Mt. 7:17-18)! The all-holy Son of God could not be the fruit of any other tree than the Immaculate Virgin.
Holy Mary, Mother of God
Luke 1:43 "And how does this happen to me, that the mother of my Lord [Kyrios] should come to me?
Kyrios - the Greek word used by the Jews in the Septuagint Bible (Greek translation) for Yahweh, the Divine Name of God. Thus, Elizabeth is actually saying: How is it that the mother of my God should come to me. Jesus is a single person, a Divine Person, the 2nd Person of the Most Holy Trinity. To be mother of the Person Jesus is to be mother of a Person who is God. The Church gave the title Mother of God to Mary in 431 AD in order to condemn opinions of that day which denied the unity of the two natures of Christ in the Person of the Word.
Pray for us sinners, now and at the hour of our death. Amen.
Luke 2:35 "...and you yourself a sword will pierce so that the thoughts of many hearts may be revealed."
John 2:5 "His mother said to the servers, "Do whatever he tells you."
Mary sees a need and appeals to Her Son to satisfy it. He does.
The Power of Intercessory Prayer:
The saints of the Old Law and the New Law, the 12 Patriarchs and the 12 Apostles, intercede for those on earth.
James 5:16b "the fervent prayer of a righteous man is very powerful."
Rev. 5:8 "When he took it, the four living creatures and the twenty-four elders fell down before the Lamb. Each of the elders held a harp and gold bowls filled with incense, which are the prayers of the holy ones.
The angels, too, mediate our prayers. This is taught explicitly in the Jewish book of Tobit (Tob. 12:12) accepted by Christians as inspired until Luther on his own authority rejected it. It remains part of the Catholic Bible.
Tobit 12:12 I can now tell you that when you, Tobit, and Sarah prayed, it was I who presented and read the record of your prayer before the Glory of the Lord; and I did the same thing when you used to bury the dead.
Rev. 8:3 "Another angel came and stood at the altar, holding a gold censer. He was given a great quantity of incense to offer, along with the prayers of all the holy ones, on the gold altar that was before the throne."
Vain repetition is mindless babbling not repeated prayer
The purpose of the different beads on the rosary is to count the various prayers as they are said. Unlike the Moslem prayer beads and the mantras of Buddhism, the prayers of the rosary are meant to occupy our whole being, body and soul, in meditating on the truths of the Faith. Simply to repeat prayers is not the vain repetition condemned by Christ (Mt 6:7), since He Himself repeats His prayer in the Garden three times (Mt 26:39, 42, 44) and the Psalms (inspired by the Holy Spirit) are often very repetitive (Ps 119 has 176 verses and Ps. 136 repeats the same phrase 26 times).
Matthew 6:7 In praying, do not babble like the pagans, who think that they will be heard because of their many words.
Praise the LORD, who is so good;
God's love endures forever;
 Praise the God of gods;
God's love endures forever;
. . .
 Praise the God of heaven,
God's love endures forever.
Matthew 26:39 He advanced a little and fell prostrate in prayer, saying, "My Father, if it is possible, let this cup pass from me; yet, not as I will, but as you will."
Matthew 26:42 Withdrawing a second time, he prayed again, "My Father, if it is not possible that this cup pass without my drinking it, your will be done!"
Matthew 26:44 He left them and withdrew again and prayed a third time, saying the same thing again.
The Church believes that it is necessary for a Christian to meditate (prayerfully think about) the will of God, the life and teachings of Jesus, the price He paid for our salvation, and so on. Unless we do this we will begin to take these great gifts for granted and ultimately fall away from the Lord. Every Christian must meditate in some way in order to preserve the gift of salvation (James 1:22-25). Many Catholic and non-Catholic Christians prayerfully read and apply Scripture to their lives - that too is meditation.
The rosary is an aid to meditation. As one prays the rosary, the hands, the lips, and to a certain extent the mind, are occupied with the Creed, the Our Father, the Hail Mary and the Glory Be. At the same time one is supposed to meditate on one of 15 mysteries, from the Annunciation through the Passion, to the Glorification. By means of the rosary we learn what makes true holiness ("let it be done to me according to thy word"), about the great gift of salvation ("It is finished!") and the great rewards God has in store for us ("He is Risen"). Even Mary's own rewards (Assumption and Glorification) anticipate and teach us about our own sharing in the reign of Christ.
The faithful recitation of the rosary according to this pattern has been found by Catholics to be the door to greater gifts of prayer and holiness, as shown by the many canonized saints who have practiced and recommended the rosary, as does the Church.