Could you please explain the pledge of allegiance in the war between Aishah , Muawwiya and Ali?

The Bai'ah (pledge of allegiance) is given to the Caliph, to recognize his authority.

When the third Caliph, Uthman Ibn Affan was murdered in his home by Muslim rebels, it led to a civil war between the Sahaba. It was the first time in history that the Caliph was murdered by Muslims and they did not know what to do.

The Sahaba split into two groups, one group elected Ali  as the next Caliph and gave him the bai'ah, while the other group refused to give him the bai'ah until he executed the murderers of Uthman.

This led to two major battles of Sahaba against Sahaba, the battle of Jamal (camel) where Ali's army fought against Aishah's army, and the much longer and bloodier battle of Siffeen where Ali's army fought against Muawiyah's army.

Muawiyah in particular led the battle against Ali because he was Uthman's closest blood relative who was in a position of authority (he was the ruler of Syria since the time of Umar) and he took it as his responsibility to bring the murderers of Uthman to justice.

The issue wasn't resolved until 6 months after the death of Ali at the hands of another rebellious group (the Khawarij); when his son Hassan handed over the Caliphate to Muawiyah.

When Ahmad Ibn Hambal was asked about what happened between Muawiyah and Ali, he recited the verse,

“That was a nation who has passed away. They shall receive the reward of what they earned and you of what you earned. And you will not be asked of what they used to do.”[1]

You can download the lecture series 'The Fitnah" by Kamal El-Mekki for an authentic detailed account, or if you wish to read, then Dr Ali As-Sallabi's 'The Biography of Ali Ibn Abi Talib' Volume 2 discusses it in detail.

References: 

[1] Surah al-Baqarah 2:134; Ibn Kathir, The Caliphate of Banu Umayyah, p. 121