There are slightly differing variations but the basic story is that a lady appears to humble, usually young children, and she wants them to pray because her son, Jesus, is hurt by the world's sins. She can withhold his wrath if they'll pray various things. Also, in some accounts, she can lead people to salvation. IOW, she has more power than Jesus, the Messiah.
In all these Marian apparitions the dominating figure is, of course, Mary, who says to go to her son, but always she's in control, she's the one that showed up in whatever location, and she can tell the Son of God what to do or not.
I've read one Orthodox view of the historical Marian apparitions which was good, even with the occasional gratuitous Orthodox swats at "the Latin church", and Protestantism. But the analysis of the seeming need for another Earth mother goddess cult is certainly interesting.Marian Apparations.
Some of the ones that have not been approved are Garabandal, Medjugorje, Necedah, Bayside, Conyers, and a few others.
After doing some personal research, I decided that Guadalupe is probably based on a legend and not a real event, but I know *something* happened at Lourdes, Fatima, and Knock. What, I'm no longer as sure as I once was.