This answer is not intended as an exhaustive history, an attempt to settle allegations or open questions, or judge the authenticity of the apparitions.
History. On June 24, 1981, six children in the town of Medjugorje, Yugoslavia (today, Bosnia-Herzegovina), began to experience phenomena which they alleged to be apparitions of the Blessed Virgin Mary. This apparition had a message of peace for the world, as well as a call to conversion, prayer and fasting. It also entrusted to the children secret messages about events to be fulfilled in the future. These "secrets," confided individually to different visionaries, have not been revealed to the public. The apparitions themselves have continued almost daily since 1981, with some of the now young adults continuing to experience them regularly (those who have not yet received all the secrets intended for them) and others not. Originally they occurred on a hilltop near the town where a large Cross commemorating the Redemption exists. They have since occurred in many other places, including the parish church, St. James, and wherever the visionaries happen to be located at the time of the apparition.
The news that Our Lady might be appearing immediately began to attract pilgrims to Medjugorje, first from the surrounding countryside, and then, despite the communist government of that day, from Europe and the whole world. These included clergy and theologians, as well as experts from the physical and medical sciences who testified to some kind of phenomenon taking place when the apparition was said to be occurring. The private judgement of these early visitors did much to bolster people's belief in the events at Medjugorje. In addition, some pilgrims reported seeing the sun spin and being able to look at it without pain or eye damage, others that their rosaries turned gold colored, still others that remarkable physical or spiritual/moral healings had taken place. All of these contributed to the fame of the alleged apparition.
Three Ecclesiastical Evaluations.
1. Mostar. The initial, informal, response of the Bishop Zanic of Mostar, in whose diocese Medjugorje is found, is said to have been favorable. However, it is alleged that comments attributed to the vision that was critical of the secular clergy and himself convinced him the visions could not be authentic. He nonetheless established a commission in 1982, comprised of theologians, scientific experts and religious superiors to investigate the Medjugorje events. Its three year study produced a vote from two members of the commission in favor of supernaturality, one that it was authentic initially but no longer so, one abstention and eleven votes that nothing supernatural was occurring there.
Reference: Letter of Bishop Zanic
2. Bishops Conference. Since the Medjugorje events had exceeded the scope of a local event, Cardinal Kuharic, President of the Yugoslavian Bishops Conference, announced in January 1987 that a national commission would be established to continue investigating. This decision had been communicated to the Holy See, which stated that it accepted the judgment of the diocesan commission under the authority of the local bishop but urged, as well, that the work be continued at the national level. The Bishops' Conference's instructions to the faithful were that pilgrimages should not be organized to Medjugorje on the basis of its being supernatural and that the Marian devotion of Catholics should be in accordance with Church teaching.
In April 1991 the following declaration was made by the Bishops' Conference of the former Yugoslavia, communist Yugoslavia having been dissolved into member states with the fall of the Soviet bloc.
The bishops, from the very beginning, have been following the events of Medjugorje through the Bishop of the diocese [Mostar], the Bishop's Commission and the Commission of the Bishops Conference of Yugoslavia on Medjugorje.
On the basis of the investigations so far it can not be affirmed that one is dealing with supernatural apparitions and revelations. [emphasis added]
However, the numerous gatherings of the faithful from different parts of the world, who come to Medjugorje, prompted both by motives of belief and various other motives, require the attention and pastoral care in the first place of the diocesan bishop and with him of the other bishops also, so that in Medjugorje and in everything connected with it a healthy devotion to the Blessed Virgin Mary may be promoted in accordance with the teaching of the Church.
For this purpose the bishops will issue specially suitable liturgical-pastoral directives. Likewise, through their Commission they will continue to keep up with and investigate the entire event in Medjugorje.
3. Holy See. The attitude of the Holy See has gone through three basic phases. The first phase was to recognize the competence of the local Ordinary (the Bishop of Mostar), and then that of the Bishops Conference, as specified in the Roman norms for alleged apparitions.
In the second phase, given the non-recognition of supernaturality of the local investigations, the Holy See answered particular pastoral questions addressed to it. This included simply repeating the conclusions of the canonical investigations, but also acknowledging that pilgrimages are taking place. A letter of the then Secretary of the Congregation for the Doctrine of the Faith, Archbishop Bertone, writing to the Bishop of St. Denis, France, is an example. Commenting on a statement by the then Bishop of Mostar that the alleged apparitions were not simply lacking evidence of supernaturality but were in fact NOT supernatural (i.e. definitively so), he stated:
The main thing I would like to point out is that the Holy See does not ordinarily take a position of its own regarding supposed supernatural phenomena as a court of first instance. As for the credibility of the "apparitions" in question, this Dicastery respects what was decided by the bishops of the former Yugoslavia in the Declaration of Zadar, April 10, 1991: "On the basis of the investigations so far, it can not be affirmed that one is dealing with supernatural apparitions and revelations." Since the division of Yugoslavia into different independent nations it would now pertain to the members of the Episcopal Conference of Bosnia-Herzegovina to eventually reopen the examination of this case, and to make any new pronouncements that might be called for.
What Bishop Peric said in his letter to the Secretary General of Famille Chretienne, declaring: "My conviction and my position is not only non constat de supernaturalitate, but likewise, constat de non supernaturalitate of the apparitions or revelations in Medjugorje", should be considered the expression of the personal conviction of the Bishop of Mostar which he has the right to express as Ordinary of the place, but which is and remains his personal opinion.
Finally, as regards pilgrimages to Medjugorje, which are conducted privately, this Congregation points out that they are permitted on condition that they are not regarded as an authentication of events still taking place and which still call for an examination by the Church. [emphasis added]
Reference: CDF, Letter to Bishop Aubrey
Ruini Commission. Finally, in a third phase the Holy See has undertaken a canonical investigation of the event. In 2010 Pope Benedict XVI established a Commission chaired by Cardinal Camillo Ruini, Cardinal Vicar Emeritus of the Diocese of Rome. By 2014 it had completed its work and referred the matter to the Congregation for the Doctrine of the Faith, and the ultimate decision of Pope Francis. This decision, supposedly part in favor (the early events) and part opposed (those after them), has never been released, nor has Pope Francis made any final decision on the supernaturality of Medjugorje. Instead, beginning in 2017, Pope Francis undertook the pastoral care of pilgrimage activities in Medjugorje by the appointment of a Special Envoy, despite his own critical comments made over the years about the alleged events there. Since then, the Envoy has permitted organized pilgrimages, previously forbidden, noting that Marian Shrines are not only places where Mary has appeared.
Conclusion. From the point of view of an ecclesiastical evaluation the status of Medjugorje has not changed since the 1991 Yugoslavian declaration. Absent a definitive judgment affirmed by the Pope, it has undertaken the pastoral measures noted above, while continuing to prohibit any affirmation of supernaturality. Specifically it has proscribed the seers from doing so, and forbidden them from propagating such affirmations through Medjugorje-related events. Claims that the “Gospa” is appearing there, or anywhere, are private affirmation lacking the credibility of the Church's sanction, and having the counter-credibility of two canonical investigations which concluded the events lacked evidence of supernaturality.
Mystical Phenomena. The presence of remarkable phenomena is for many sufficient evidence of the validity of an alleged apparition. For others the judgment by local Church authority that there is no evidence of supernaturality at a site suggests fraud, mental illness or the demonic. The Church for her part, however, takes great care before affirming the certain supernaturality or non-supernaturality of phenomena, as the Roman statements given above show. There are likewise few examples of outright condemnation. When they do occur it is usually on the basis of doctrine which is contrary to the faith.
The reasons for such caution are rooted in the Church's common teaching. St. Thomas Aquinas and St. John of the Cross both assert that as a general rule mystical phenomena (whether in the lives of saints or in apparitions) are the work of the angels. Unless God Himself needs to act to immediately produce an effect (such as to create out of nothing or to infuse sanctifying grace into the soul), He works through creaturely instruments. Thus the intellectual lights granted in contemplative prayer, the visions and locutions of private revelations, the levitations of the saints, the ecstasies of mystics and visionaries, and most external phenomena associated with mysticism, are produced by the angelic nature. Since both good and evil spirits possess the angelic nature the presence of such phenomena alone is an equivocal sign of authenticity. This means that a great deal of unexplained phenomena can occur without indicating positively that the event is from God. This is why the Church looks, among other things, for evident supernaturality, that is, for effects beyond the ability of men or angels which can be attributed to God alone.
Theologians remain divided in judging which phenomena fall clearly into the category of strict supernaturality. However, the practice of the Church in the canonization process of recognizing as miraculous those cures which meet certain strict criteria is a standard that has been applied in approving apparitions, as well (e.g. Lourdes, Beauraing, Banneux). At Fátima the Miracle of Sun likewise fell into the category of a natural prodigy. It is clear, however, that the phenomena which many laity have experienced in connection with alleged apparitions in our days, and which they consider to be proof that they are authentic, do not in fact rise to the level of evident supernaturality. Angelic or demonic activity would be sufficient to explain them. Without a proof of the supernatural order there is little likelihood of the Church affirming an apparition as authentic.
In the case of Medjugorje the commissions found that nothing directly connected with the apparition met this strict standard. As the earlier quoted statements show, the Church remains open to new evidence of supernaturality should it occur and has not judged that Medjugorje is NOT supernatural, much less condemned it.
What the Church has forbidden. From the statements given to date by ecclesiastical authorities it is clear that no one holding an office in the Church (bishop, pastor, rector, chaplain or other) may by virtue of that office lend official sanction to activities which tend to assert the supernaturality of Medjugorje, that is, to contradict the decisions made by competent local authority. Those statements speak only of pilgrimages organized under official auspices; however, common sense tells us that a conference or other activity sponsored by a diocese, parish or other Catholic institution would also be prohibited. Likewise, there could not be public veneration (cultus) of the Blessed Virgin under the title of Our Lady of Medjugorje, since this would suggest the certainty of her appearing there. The title Queen of Peace, however, is already part of the patrimony of the Church.
The Yugoslavian statement speaks of liturgical-pastoral directives which may be developed. Catholics would be obliged to obey whatever positive or negative directives the Bishops' Conference or the local bishop issued regarding the site.
Do the decisions of the Church amount to an obligation to believe in the intellect that Medjugorje is not supernatural? The answer is no. First, even private revelations approved by Rome bind the faithful to accept them only based upon reasonableness, not faith. Pope Benedict XIV stated,
Although an assent of Catholic faith may not be given to revelations thus approved, still, an assent of human faith, made according to the rules of prudence, is due them; for according to these rules such revelations are probable and worthy of pious credence. [Benedict XIV, De Serv. Dei Beatif.]
This means that once a private revelation has achieved Papal approbation it is unreasonable, i.e. imprudent but not against the faith, to not accept it as authentic. The contrary would also be true. If Rome judged a private revelation to not be supernatural, the reasonable person would be satisfied with that conclusion. Would they sin if they did not accept it? They might sin by imprudence, rash judgement or the like, but not against the faith or the obedience they owed the Holy Father. Catholics must always, however, following the external precepts imposed by the Church in such matters, that is, what they may or may not do, as opposed to what they think.
As far as theological judgements made at the local level, therefore, the standard could not be any higher, and is certainly lower. The issue of Medjugorje, therefore, cannot be resolved solely on the basis of the local Church's finding that there is no evidence to date of supernaturality. This is even more clear in light of the statement of Archbishop Bertone that the Bishop of Mostar's 1998 statement that it is certainly "not supernatural" is his own personal opinion. Others are therefore entitled to their personal opinions, also.
What the Church permits. As the already cited statements note, Catholics may go to Medjugorje. Such pilgrimages may even include priests acting as chaplains, as opposed to officially sponsoring them. Also, the Church has not suppressed discussion of Medjugorje, therefore, it is allowed. Common sense, however, says that Catholics on both sides of the Medjugorje issue should exercise prudence and charity in speaking of others who believe differently. Medjugorje is not a litmus test of orthodoxy, though every Catholic will have a moral obligation to accept the judgement of Rome, in the manner Pope Benedict explained, should it ever be rendered.
St. Augustine probably gave the simpliest and most helpful rule for all matters of the Church's life when he said (in my paraphrase):
In necessary things unity,
in undecided things freedom,
and in all things charity.