Abbas Effendi, who adopted the name Abdu'l-Baha ("Slave of Baha"), was the eldest son of Baha'u'llah. His father's will specified that Abdu'l-Baha was to be his successor. However, the will also explicitly states that anyone who claims to be the next Manifestation within the next thousands years is a liar.
This put Abdu'l-Baha in the position of being an infallible representative but without the same kind of divine authority held by his father. He also came into conflict with his younger half-brother Mirza Mohammed Ali, whom Baha'u'llah had named his successor after Abdu'l-Baha.
Center of The Covenant:
Abdu'l-Baha took the title "Center of the Covenant," implying that he was link between God and humanity. This was a position many Baha'is felt could only be filled by a Manifestation of God, of which Abdu'l-Baha could not be. Abdu'l-Baha also claimed that only he had the right to interpret his father's writings, and that his interpretations were infallible.
While he did not explicitly claim it, many loyal Baha'is came to view Abdu'l-Baha as an extension of his father, so while he was not a Manifestation of God in his own right he was still generally viewed as a divine being (as all Manifestations are).
Split Between Brothers:
Ali took the lead among objectors to Abdu'l-Baha's claims of authority. Many other family members and old and trusted friends of Baha'u'llah also voiced dissent. Abdu'l-Baha's response was to excommunicate them and label them "Violators of the Covenant."
In 1901, the strife between the two feuding sides became so intense that the Ottoman Empire confined the Baha'is once more to Akka, revoking the freedom to travel that they had experienced for many previous years. This continued until 1908, when all political prisoners – of which the Baha'is were included – were freed as part of the Turkish Revolution.
Unlike his predecessors, Abdu'l-Baha traveled extensively, including Egypt, Great Britain, France, Germany, Hungary and the United States. These travels were missionary in purpose, exposing greater numbers of people to Baha'i teachings as well as setting an example for them of how Baha'is are expected to conduct themselves.
In 1912, Abdu'l-Baha was present for the dedication of the Baha'i temple in Chicago, one of only seven Baha'i temples in the entire world. The temple would not be finished for more than forty years.