In 1531, a poor Aztec named Juan Diego traveled through Tepayac hill country. He ran into a beautiful woman surrounded by light. She identified herself as "Mary, mother of the true God", and she told him she desired a church be built in that location.
Juan, imitating Mary's own haste when she was called to help her cousin Elizabeth, flew to the palace of the Bishop in Tenochtitlan and told his story. The Bishop-elect, Fray Juan de Zumarraga, listened to Juan's tale and said he would think about it.
As proof of her appearance, Mary told Juan to pick flowers from the top of a freezing hillside. The flowers were there, and Juan picked them and took them to the Bishop as proof, since flowers shouldn't have been growing at that time of year. When he opened his tilma, a poncho-like cape made of cactus fibers, the image of Our Lady was imprinted on the fabric.
That's the short version.
Where did her title come from? She requested it, though it's believed that she actually said Santa Maria de Coatlallope, which means one who treads on snakes. The local Aztec dialect caused the mis-translation.
As for miracles and cool stuff associated with the image?
First, the tilma's very existence. It has maintained its structural integrity over nearly 500 years, while replicas normally last only 15 years before suffering degradation. The image's longevity (cactus fibers should have disintegrated by now) is inexplicable, as is the method used to create the image, though it's been studied by experts.
Next, it repaired itself after suffering considerable damage from an ammonia spill in 1791.
Then, on November 14, 1921, a bomb damaged the altar but left the icon unharmed.
The next one needs a more technical explanation, and the Wikipedia description is very good.
In 2012, the St. Augustine Chapel, which displayed a portrait of the image, burned to the ground. The image survived with only minor damage.
There's only one thing left to say:
Our Lady of Guadelupe, pray for us, because we really, really need it right now.