Why are there differences of opinion about divorce?

2:228 Divorced women remain in waiting for three periods...and their husbands have more right to take them back in this [period] if they want reconciliation...
2:229 Divorce is twice. Then, either keep [her] in an acceptable manner or release [her] with good treatment...

As the above excerpts from the Holy Qur'an indicate, a divorce (i.e., the talaq) is immediately followed by the waiting period (i.e., the iddat), during which time the husband has the right to revoke the divorce and take his wife back. However, this process can only be done twice; the third divorce is considered permanent no matter what.

65:2 And when they have [nearly] fulfilled their term, either retain them according to acceptable terms or part with them according to acceptable terms. And bring to witness two just men from among you and establish the testimony for [the acceptance of] Allah .

At the end of the waiting period, the couple must choose either to reconcile or to make their divorce permanent. This decision is to be witnessed so there is no question as to the resolution. There is some question as to whether the original talaq needs to be witnessed as well, but without any evidences to mandate it witnesses shouldn't be considered a requirement.

2:230 And if he has divorced her [for the third time], then she is not lawful to him afterward until [after] she marries a husband other than him.

After the divorce is rendered permanent, the couple are explicitly forbidden from remarrying each other until the woman marries (and subsequently divorces) another husband. The rite of halala, a form of temporary marriage wherein a man marries and divorces a woman only to make her lawful to her ex-husband, is to be avoided; according to the following hadith, such marriages are clearly against the sunnah of the prophet:

The Prophet (peace\_be\_upon\_him) said: Curse be upon the one who marries a divorced woman with the intention of making her lawful for her former husband and upon the one for whom she is made lawful. (Sunan Abu-Dawud 11:2071)

There is also some dispute whether the three divorces can be performed all at once (i.e., the triple-talaq), which would instantly make the divorce irrevokable. The crux of this dispute can be found in the following hadith:

Abu al-Sahba' said to Ibn 'Abbas: Enlighten us with your information whether the three divorces (pronounced at one and the same time) were not treated as one during the lifetime of Allah's Messenger (may peace be upon him) and Abu Bakr. He said: It was in fact so, but when during the caliphate of 'Umar (Allah be pleased with him) people began to pronounce divorce frequently, he allowed them to do so (to treat pronouncements of three divorces in a single breath as one). (Sahih Muslim, 9:3493)

While there are scholars who follow the example set by Umar (treating the triple-talaq as irrevokable), the prevalent opinion is that triple-talaq should be treated as a single talaq, as that was the sunnah of the prophet. In this case, a second or third talaq would only be counted after the wife had been taken back from the previous iddat.