She wanted her cousin Talha to succeed Uthman
(The first Umayyad caliph).
She was jealous of Ali.
These have been demonstrated by historical evidences collected by Al-Shaykh Al-Mufid. An example is the narration is that once she heard the news of murdering of Ali ibn Abi Talib in mosque, she gratefully prostrated in happiness.
(This is a summary translated from the links in the end. If you need more details and more references for each part, let me know or ask as a separate question)
Shia and Sunni scholars have different views about the causes of the war mostly because Aisha is highly regarded in Sunni view. However, although Aisha is respected by Shias and insulting her has been made forbidden, she is not regarded as a high ranking sahaba (prophet's companion) and is highly criticized for her conduct especially after the passing of Prophet Muhammad, peace be upon him, and his family and righteous companions.
The Shiite narrative is that, Aisha, the daughter of Abu Baker from Bani Tamim tribe, treated Uthman differently over two periods. In the beginning she supported him along other Muslims. But after a while, when Uthman distributed power among his close Umayyad relatives, muslim public went angry. During Umar’s Caliphate, Aisha used to get 12,000 Dirhams in governmental salary each month. But when Uthman succeed Umar conflicts grew between him and Aisha, and he finally cut her salary. Aisha among other notable Islamic figures (such as Talha and Zubair) declared Uthman as Kafir (apostate). Emboldened by the Prophet's wife open opposition to the caliph, the already dissident Muslims got together and seized Uthman’s house. Uthman access to water was prevented but Ali ibn Abi Talib and his sons protected Uthman’s home and delivered water to Uthman. Their reason for supporting Uthman at the time was that they wanted to prevent the precedent of killing of caliphs by the Ummah. But despite their efforts, Uthman was ultimately killed by revengeful muslims and his dead body remained on ground for 3 days and no one did any funeral for him. Finally Muslims left his body in rubbish wasteland outside the city where became his grave. But years later when Muaviyeh declared himself the caliph he extended the cemetery of Medina to include the grave of Uthman.
When Imam Ali (from Bani Hashim descent) accepted Caliphate in response to Medina muslims’ request, Muaviyeh, who was then the governor of Sham (Syria), sent two letters to Talha and Zubair and called them "Amir ul-Mumenin" (commander of the faithful) and told them that the people of Basra and Kufa wanted them to be their governors but they should rush before Ali’s governors take over the cities. Talha and Zubair came to Ali and asked him to give them the governments of Kufa and Basra arguing that they were the ones who helped Ali to rise to power by contributing to the fall of Uthman. But Ali refused their requests telling them that he needed them the most in the capital, Medina. His refusal also implied that he didn't consider them qualified for the positions. Upset with Ali’s refusal, Talha and Zubair went to Mecca under the pretext of pilgrimage to prepare an army to occupy Kufa and Basra by the support of Muawiya’s associates.
Aisha, who was in Mecca at the time, was delighted upon hearing the news of murder of Uthman and decided to travel to Medina to see her tribesman (from Bani Tamim tribe) succeeding Uthman. But on her way (to Medina) at Serf caravansary, she met Ibn Umm Kilab who told her that the people of Kufa had pleaded allegiance to Ali as their Caliph. Aisha, furious at the news, went back to Mecca and declared that she was to take revenge on the killers of Uthman (meaning people of Medina who had pleaded allegiance to Ali). Ummu Salma another wife of the prophet refused to accompany Aisha and even tried to discourage her by reminding her of prophet’s hadiths stressing the superior status of his son-in-law, Ali ibn Abi Talib over the rest of Muslims. Other among prophet’s wives also discouraged her and supported Ali’s caliphate. But Aisha made up other excuses to carry on with the rebellion. She allied herself to Talha. He and Zubair relied on her status as prophet’s wife to legitimize and garner support for their rebellion against Ali. Thus Mecca turned into the center of those who opposed Ali Ibn Abi Talib, for various reasons. Uthman's governors financed the war against Ali. Among them were: Abdullah Ibn Rabi’ah governor of Sana, Yemen; Ya'li Ibn Umayyah, one of commanders of Uthman, and others.
On their way to Basra, The army of Mecca reached a place called Hawab where they heard dogs barking. Aisha immediately recalled a prophecy by Prophet Muhammad in which he related to them that he saw one of his wives passing over a location named Hawab while dogs barking at her, and told them lest they be the one finding themselves in the situation. Upon the recalling of the alarming prophecy, Aaisha changed her mind and decided to return to Mecca but Abdullah Ibn Zubair testified falsely that they had already passed Hawab behind long ago, and Aisha was, thus, fooled.
Ultimately the army arrived at Basra and secured it after suppressing opposing groups who had recognized the ill intentions behind the rebellion against the established Caliph. They put the governor of Basra who was faithful to Ali under arrest. Some rebels plundered the city treasury after killing the guards.
Ali Ibn Abi Talib, departed Medina to stop the rebellious army. His envoys finally managed to encourage ten thousands from Kufa to join the Medina army of four thousands. Notable figures among Ali’s army were Ibn Abbas, Ammar Yasir, Haatam Ta’ei, Malik Ashtar, Muhammad Ibn Hanafiya, Muhammad Ibn Abi Bakr (Aisha’s brother) and Ali’s sons, Hassand and Hussein.
Ali sent many messengers to the three rebel leaders (Aisha, Talha and Zubair) to discourage them from war and to invite them to unity but despite the strong arguments of his messengers the rebel leaders were determined to fight as they thought they can easily win the battle.
The war started while Aisha was seated on a camel and dozens protecting her ride. She and her camel were seen by the rebelling army as a testimony to their legitimacy and also a source of morale as she was viewed as the wife of the holy Prophet. During the war Ali made many attempts to discourage the rebels and its leaders from war but to no avail. As war continued Ali ordered his men to approach the camel carrying Aisha and severe its legs. The camel collapsed and Aisha surrendered herself and the battle was over. Talha and Zubair were killed in the meantime.
Despite Aisha’s main role in agitating people against Ali Ibn Abi Talib, he treated her with utmost respect and ordered her brother and some women disguised in men’s dress to accompany her to Mecca. Aisha on numerous occasions expressed much regret and remorse at her decision to rebel against Ali.
Ali also felt much pity for the ill fate of Talha and Zubair who used to be among faithful supporters of the Holy Prophet during his life. He recited prayers for all victims of the war including those of the rebels. The popular estimate sets the total death toll of the war at 20,000.
In the aftermath, Ali ordered all properties of people of Basra taken by his army returned to them unless those with government asset marks. He also refrained from suing those involved in the rebellion.
During his lifetime, Prophet had predicted that Ali would have to fight with three groups, Nakesin, Qasetin and Mareqin. His companions were always interested to know these three groups were. “The Battle of the Camel” was the realization of Nakesin’s war against Ali.
www.hawzah.net (107 historical references are mentioned at end of this article)