A Marian apparition is a supernatural appearance of Mary to a person (or group of people) on earth. Since the early centuries of the Church, there have been thousands of reported Marian apparitions. “Reported” is a key word here; just because someone says that they saw Mary doesn’t mean that they really did. The Church, under the guidance of the bishop in whose diocese the alleged apparition occurs, carefully investigates alleged apparitions according to a set of criteria established by the Sacred Congregation for the Doctrine of the Faith.
It’s worth pointing out that the Church never judges an apparition to be “authentic”; the strongest positive statement that they will make is that an apparition is “worthy of belief.” Even the small handful of apparitions judged “worthy of belief” (which include Mary’s appearances at Lourdes, France, and the visits of Our Lady of Guadalupe in Mexico) are considered to be “private revelation,” and are therefore not required elements of belief. Many Catholics have a strong affinity for certain Marian apparitions; other Catholics never give them much thought.
And why would Mary bother to come to earth in the first place? Well, the character of her visits offers a clue. Many of her messages carry the same themes: a call to prayer, a desire for peace, an invitation to personal penance. I personally like to think of Marian apparitions as ways that she gets our attention when we’ve strayed off course, nudging us back on the path that was laid out for us by Christ.