Well, we Catholics actually believe in the priesthood of every baptized person. As long as the person, at any age even if they are too little to understand it yet, has been baptized in the Name of the Father, and the Son and the Holy Spirit by a ministerial priest (or any other person in case of emergency), that person becomes a prophet, a king and a priest. The person can also get baptized by desire if it can’t get baptized or by martyrdom. The priesthood of the ministerial priests (the ones that can celebrate Mass and administer the Sacraments) is different than the royal priesthood every baptized shares.
Sorry for going off topic, but felt the need to point that out. Now, on topic stuff: We believe in the prayer to the Blessed Virgin Mary because we really see her as our true Mother and the most powerful intercessor (even if Jesus is the only mediator between God and man, she prays for us just as you can pray for your beloved ones, so we ask for her intercession), since she being the Mother of God, her Son can’t deny anything she asks for. Jesus on the Cross presented her to John as his Mother, and in that sign He presents her to all of us as our Mother. As our Mother she loves all of us and would do anything to have our souls saved and help us avoid sin. Also, she is mediator for all graces, and has claimed that she will not deny anything that is asked to her when praying the Holy Rosary (as long as it’s not something that goes against the Will of God, of course; this was a promise of Rosary she left in an apparition).
Of course, we don’t worship her, as we clearly know she is not God and is no greater than her Son Jesus Christ, but we venerate her as the Queen of all saints, and know that every prayer we do to her is also done to her Son.
She is also one of the greatest mysteries of the Catholic faith, as she apparently chose that the apostles and preachers of her time should not talk about her but only tell the necessary and most essential stuff, so that’s why the Bible tells almost NOTHING about her, However, she is mentioned since the first book of the Old Testament (she is the woman who God says will step on the serpent’s head) and is mentioned in the Apocalypse, too (she is the woman robed with the sun, with the moon under her feet and a crown of twelve stars). Everything we currently know about Mary has been slowly discovered with the light of the Holy Spirit through the centuries, but due to the highness she was elevated to, we will never know everything about her.
There’s a lot that can be said about her, the most perfect creature of all: She was conceived without the original sin by the omnipotence of God (this is called the dogma of the Immaculate Conception), stayed a virgin all her life and never met sin, and this was because the Mother of the Christ had to be a totally pure creature. There’s a lot that you can investigate about her in official Catholic media, many saints had written about her (some of the most recommended readings are “True Devotion to the Blessed Virgin Mary” by St. Louis-Marie Grignion de Montfort and “The Glories of Mary” by St. Alphonsus Maria de Liguori). There has also been multiple apparitions of her through history, always leaving a message, and there’s even an image of herself that she left in one of the apparitions and no man painted it, Our Lady of Guadalupe who appeared in Mexico (the image she left is of her but with Mexican facial features and skin color, and that image has been studied several times by scientists and found out to be something impossible for a human being to paint it the way it was at the time she appeared). Get to truly know her and you will fall in love. She, together with Jesus, is my best friend. Sorry for the lengthy reply. Blessings!