Short answers, pulled from the story.
A dispensation in Catholic canon law is an exemption from the immediate obligation of a law in a specific case. Its purpose is to modify hardship caused by the rigid application of general laws to particular circumstances, while leaving the law itself intact for all other cases. A dispensation requires a just and reasonable cause and ceases to exist when that cause no longer applies.
The power to dispense belongs to the original lawgiver, his successors or superiors, and those to whom the right has been delegated. Because no authority stands above the pope, he can dispense from any canonical law. Bishops can dispense from impediments not reserved to the pope, and vicars-general hold the bishop's ordinary dispensing powers by virtue of their appointment.
The Council of Trent left the medieval theory of dispensation intact while guarding against abuses. It rejected a proposal to subject papal dispensing power to approval by a general council, and decreed that dispensations could only be granted for just and urgent causes or for a clear benefit to the Church. Trent also ruled that payment of money for a dispensation would void it entirely.
A matrimonial dispensation is the relaxation of an impediment that would otherwise prohibit or annul a marriage, granted in a particular case. It can be issued before a marriage takes place or to legitimize one already contracted. Matrimonial dispensations can operate in the internal forum, concerning conscience, or the external public forum, or both.
Pope Pius X reorganized the Roman Curia through the apostolic constitution Sapienti, which took effect on the 3rd of November 1908. The newly established Congregatio de Disciplina Sacramentorum took over dispensing power for poor and quasi-poor petitioners in public cases, while the Penitentiaria was limited to secret impediments in the internal forum. The Congregation of Extraordinary Ecclesiastical Affairs lost all its dispensing powers.
The Council of Trent decreed that dispensations must be free of charges, but the Roman Curia permits four categories of legitimate costs: carriage and agent fees, an administrative tax, a componendum directed to pious uses, and an alms the petitioners distribute themselves. Diocesan chanceries may only collect the modest fees established by the Innocentian Tax, approved by Innocent XI on the 8th of October 1678. Neither the componendum nor the alms goes to the pope or curia members personally.