Imam Ali (A) was appointed as the Prophet's successor in Ghadir Khumm, but some of the companions opposed him due to some reasons that are going to be mentioned.
The Prophet (P) appointed Imam Ali (A) as his successor in Ghadir Khum while addressing people who returned from Haj. Many companions and several original sources have confirmed the event. [1] But the question is that why some prominent companions refused to obey the prophet (P) misleading other Muslims who follow them. Here are some reasons: First, some distinguished companions used to oppose the Political decisions of the Prophet (P) as Umar opposed him in Hudaybeyah, Battle of Hunain, etc. that is why he deprived Imam Ali (A) of his divine right. Second, Imam Ali (A) was young so based on Arab tradition; he was not eligible for the position, despite being already appointed by the prophet. Third, Imam Ali (A) was so closed to the Prophet (P) and it made some companions jealous of him. Also, Imam Ali (A) had many characteristics that they did not have, for instance, he was the Prophet’s cousin and son-in-law. Moreover, many Qurayshids, as a prominent Arab tribe, had a deep enmity towards Imam Ali (A) for he killed many of their grandees in the Battle of Badr, Uhud, Trench, etc.
[1] 110 companions, 83 Tabiun (followers) and 360 Sunni scholars have narrated the event. See: https://en.wikipedia.org/wiki/The_event_of_Ghadir_Khumm#Ghadir_Sermon