Short Answer
A husband who wants to divorce his wife should say, “I divorce you” or “You are divorced”. This results in a revocable divorce pronouncement and his wife begins her waiting period [ʿiddah] of three menstrual cycles. He may revoke this divorce by saying ‘I take you back’ or ‘I revoke the divorce’. If he is intimate with his wife by kissing or caressing her with desire then the divorce is also revoked automatically. If the divorce is not revoked within the waiting period [ʿiddah] it becomes final and the two are no longer husband and wife.
A woman who wants to divorce her husband should ask him for a release [khul`] from the marriage. She may offer him some money, since he actually gave her some money [mahr] when they got married. If the problem in the marriage is from his side then he should not take any money. If the problem is from her side then he should not take more than he had given her as dowry [mahr].
If the husband refuses to grant his wife a release [khul`] or demands an unfair amount of money for that, the woman should petition an Islamic tribunal consisting of three scholars explaining that he husband is unfairly keeping the wife stuck in marriage with no valid reason. They evaluate the case and then either force him to accept a fair release [khul`] or, if unable, to grant a faskh [annulment] of the marriage which results in them no longer being husband and wife.
Advice
It is very important to get third parties such as family members and counselors involved to resolve disputes between husband and wife before considering divorce, especially when the couple has children. However, marriage is not a death sentence and it is allowed in Islam to end a marriage when there are good reasons for doing so. No one should rush to get out of a marriage for flimsy reasons nor should anyone be pressured to remain in a highly problematic marriage.
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Shaykh Mustafa Umar