Each year thousands of women in Pakistan separate from their husband through Khula, this peaceful exit from marriage contract was unknown to the world when Islam gave women her basic rights.
The Shariah evidence for Khula is based on the incident of Thaabit ibn Qays, when his wife came to Prophet Muhammad (PBUH) and asked for divorce.
It is narrated by Bukhari that ‘The wife of Thabit bin Qais came to the Prophet (?) and said, “O Allah’s Messenger (?)! I do not blame Thabit for defects in his character or his religion, but I, being a Muslim, dislike to behave in un-Islamic manner (if I remain with him).” On that Allah’s Messenger (?) said (to her), “Will you give back the garden which your husband has given you (as Mahr)?” She said, “Yes.” Then the Prophet (?) said to Thabit, “O Thabit! Accept your garden, and divorce her once.” (Sahih al-Bukhari 5273)
From this scholars deduce that if a woman is not satisfied with her husband then she can ask her husband for divorce. If the husband is not willing to give divorce then the wife can seek justice from Islamic courts which can ask the husband to divorce the wife, if he denies then the qadhi (Judge) can dissolve the marriage.
One of the important aspects of khula is that the wife loses her right to the mahr she received at the time of marriage. She can only keep it if the husband gives her the right otherwise she has to return it.
To conclude the main difference between Khula and Talaq is that Khula is a demand of divorce by wife while Talaq is initiated by the husband on his own will.
After iddat the divorced woman is free to remarry again according to Islam. Prior to Islam women had no right to divorce while men were free to give divorce on any basis and marry as many women as they can.
It was Islam that put conditions for marriage contract, outlined the duties of husband and wife, defined their roles and gave woman the right for khula.