The division of Baha'is into several different sects came about because of the differing interpretations of the Will and Testament of Abdu'l-Baha. In that holy and sacred Document, Abdu'l-Baha created the Institution of the Guardianship as the Center of the Cause, and the Institution that would carry forward His function and authority as Interpreter of the Holy Writings. He appointed Shoghi Effendi as the first Guardian of the Cause and enjoined upon him the duty to appoint his successor: either his first born descendant, or the Guardian may "choose another branch to succeed him."
Since Shoghi Effendi had no descendants, the question came up whether he chose another branch. The Arabic word, Aghsán, or Branch, had been defined by Shoghi Effendi as the Sons of Baha'u'llah, although this was obviously not the meaning of the word in this context, since at the time Abdu'l-Baha wrote these words, there were no faithful Sons of Baha'u'llah. The proper interpretation of the word Aghsán to the question of the continuing Guardianship is key to understanding the split amongst the Baha'is.
There are three basic approaches to this:
1) Magic Sperm: The first approach is the belief that only one of the Sons of Baha'u'llah, or at least a male blood-line descendant, is eligible for appointment as Guardian. This is the majority Baha'i view represented by those Bahais who presently occupy Haifa and Wilmette (the heterodox Baha'is). I consider this concept to be the "magic sperm" theory carrying the implicit belief that there is some magical essence in the bloodline of Baha'u'llah that is necessary for one to be divinely guided. Although this is a highly superstitious and materialistic point of view, there is an historical basis for this in Islamic belief. For example, the descendants of the Prophet Muhammad are considered to have some special status or respect. The idea of blood royalty found in many of the systems of government historically also carry this idea that the genetic code of certain families carries with it some kind of magical power or essence.
If we carry the logic of this belief to its extreme, we would consider what would happen if the genetic material of Shoghi Effendi were cloned. We would have an exact duplicate of Shoghi Effendi's genetic makeup, but of course physically we would end up with his cloned identical twin. It would be a different person entirely and would not be the Guardian. Of course, if we could do this, we could clone Baha'u'llah Himself, but of course we would end up with a cloned human being and not the Manifestation of God for this Day. There is no magic in their genetic material. This example does not intend any disrespect, but merely to show absurd it is to insist that only a physical relative of Baha'u'llah could possibly serve as Guardian.
2) Spiritual Aghsán: The second approach is the belief that the Guardian may choose any male believer as his successor, and that the person so chosen is, effectively, a "spiritual" Aghsán. This would be the view represented by the Orthodox Baha'is, that is those who recognize Joel B. Marangella as the third Guardian. It is also the view taken by those Baha'is who follow Jacques Soghomonian as their fourth Guardian. This approach finds that the Will and Testament did not impose an obligation upon the Guardian to appoint an Aghsán, but allows him to appoint any male believer as his successor. Branchhood is essentially a spiritual relationship and does not require a physical family relation. This is not a materialistic and superstitious approach, but rather is a spiritual understanding of the essential relationships of the Institutions of the Faith and the believers. For more on these spiritual relationships, see Joel B. Marangella's article entitled "Tree of the Covenant".
3) Secular Adoption: The other approach is the belief that the Guardian may only choose an Aghsán to be his successor but that by a legal adoption, recognized by secular or civil authorities, of any male as his son will validate that choice as being legally an Aghsán. This is the position taken by the Baha'is Under the Provisions of the Covenant. In my opinion, this approach is very similar to the second one, but curiously adds the unnecessary requirement that the Guardian's choice of successor have a legal sanction by a secular adoption.
It is ironic that the split of the Faith comes over the interpretation of a clause contained in the Will and Testament of Abdu'l-Baha, a writing intended by Him to avoid a split within the Cause of God by creating the Institution of the Guardianship and enjoining all of the believers to be obedient to the Guardian and his interpretations. Unfortunately, the vast majority of Baha'is lost faith in the Covenant of Baha'u'llah and rejected the Guardianship following the death of Shoghi Effendi, and thus they adopted the materialistic and superstitious belief that Shoghi Effendi was unable to appoint a successor because the Aghsán were either dead or unfaithful. Having failed to embrace the living Guardian of the Cause, the Baha'is followed the incorrect interpretation of the Will and Testament and have found themselves in grievous error.