Hail Mary!
Full of grace,
the Lord, is with thee!
Blessed art thou, among women!
And Blessed, is the fruit of thy womb, Jesus!
In this first part of the prayer, this is just repeating a part of the New testament....the scene is Elizabeth, Mary's cousin and also, the Mother to be, of John the Baptist. This is what Elizabeth says to Mary when she greets her....they BOTH are pregnant at this meeting up.... this is where the Bible states that Elizabeth's baby in womb, leaps for joy in the presence of the Lord Jesus, who is in Mary's womb....
So this part of the prayer comes directly out of verses in the Bible being repeated.
Holy Mary, mother of God,
Pray for us sinners, now and at the hour of our death. Amen.
Catholics, and all Christians for the most part, consider Jesus as a manifestation of God in the flesh, so Mary is the mother of God, (I suppose?)
And the rest is not asking her to forgive you herself, or that she can offer her own grace for forgiveness of sins, or that she can perform some miracle for you, or anything at all like that... She is considered as someone who is in Heaven and by her Son's side...and KNOWN as someone who has Jesus's ear...known as someone who can plead, (intercede) with Jesus, to help...
which is also supported in the new testament, with the wedding feast scene, where she finds out that they are out of wine, and then let's Jesus know that they are out of wine, and Jesus performs his very first miracle in the Bible,(asked of Him by Mary), and changes water in jars, to wine, so that the wedding feast could continue....without wine, the wedding feast would end....
So Catholics believe the dead saints, and prophets, Moses, and others are in Heaven with God, and that they are ALIVE and well in Heaven, that there is an "after life" and some lucky saints are immediately brought to Heaven when they die, no waiting for judgement day.
And Mary, along with the Saints, and the Prophets, are there to intercede and interact with Jesus/ God in person. No different than asking a congregation to pray for one of their sick members....when you say to someone, please pray for my father, he has cancer, yahdahdahdahdahdah. There is no difference to Catholics in asking that congregation to send prayers for you, than asking the saints, and prophets to send prayers for you.
In general, people pray to God first, even Catholics....praying to these others seems to be some sort of insurance....hahahahaha....just in case God was really busy at the time, these other saints and prophets can give it a second, third, fourth, fith etc shot at it for you.... and maybe get God's ear?
Personally, I just go straight to God....but this is what I read about your question on the Catholic Church's internet question and answer site.... on the subject, a while back....