What is the Hail, Holy Queen prayer?

The Hail, Holy Queen (known as the “Salve Regina” in the original Latin) is one of the most well-known prayers about Mary. It is a prayer of petition, asking Mary for her prayers on our behalf. Various sources place its origins in the eleventh or twelfth century. In medieval times, monks and friars sang it at the end of the day, and it is currently one of the four prayers to Mary that make up a part of the Divine Office (the official prayers of the Church that are prayed at certain times of the day). The Hail, Holy Queen is also prayed at the conclusion of the rosary.
The prayer is:

Hail Holy Queen, Mother of mercy, our life, our sweetness, and our hope. To you do we cry, poor banished children of Eve. To you do we send up our sighs, mourning and weeping in this valley (some use the word “Vale” here as well) of tears. Turn then, most gracious advocate, your eyes of mercy towards us; and after this our exile, show unto us the blessed fruit of your womb, Jesus. O clement, O loving, O sweet Virgin Mary. Pray for us, O Holy Mother of God, that we may be made worthy of the promises of Christ. Amen.