As Catholics, we believe that Our Lord Jesus Christ and His mother, the Blessed Virgin Mary, have regularly appeared to various saints throughout the world. Many of those apparitions have been approved by the Church and continue to feed and strengthen the faith in Our Lord throughout the world.
Sadly, there are also supposed apparitions which have been either discouraged or outright discredited. Recently, one such "apparition" was declared by the bishop of Cleveland, OH, to be without supernatural origin. In response, the "seer" of this "apparition" (and yes, I'm putting them in scare quotes on purpose) published a message which speaks of disobeying the local ordinary.
If you want to know whether an apparition is valid or not, look at how the visionaries respond to criticism from their bishops. In every legitimate apparition where this has occurred, Our Lord and the Blessed Virgin Mary have counseled the visionary to remain faithful and obedient to the legitimate Church authority. In fact, obedience to local ecclesiastical authority is a requirement for approval of an apparition. It becomes 100% crystal clear that this is a fraud when a "visionary" reports that he or she has been told by Our Lord or Mary to disobey the bishop or pastor. Our Lord would never ask for disobedience against the authority of bishops which comes from Him, and Mary would never go against the will of her Son.
As a priest, my advice would be to avoid any apparitions which have been disapproved by their local ordinary, even if there's some question to whether the bishop's decision was the appropriate one. Devotion to approved apparitions is a good and laudable action, and I would highly encourage all Catholics to learn about the legitimate apparitions, such as Our Lady in Lourdes and Fatima, the Our Lady of Guadalupe image, and the Divine Mercy. In these apparitions, among many others, Our Lord and Mary encouraged the visionaries to obedience. May we be willing to be obedient as well.
Update: Patrick Madrid answers on EWTN's Open Line radio program regarding the "good fruits" at Medjugorje.
Update 2: More on Medjugorje: Note the position of the Congregation for the Doctrine of the Faith in the last three paragraphs.