If a married couple gets divorced, under what circumstances would it be permissible for them to get married again?
The Torah forbids the couple from remarrying each other if the woman had married somebody else in the interim. Meaning, if a couple married and then divorced, and after the divorce the woman married somebody else, she may not remarry her first husband after her second husband dies or divorces her. This applies even if she and the second husband only performed Kiddushin (betrothal), that is, he gave her a ring and announced that she is betrothed to him with that ring. Even if they get divorced immediately after Kiddushin, before consummating the marriage by living together as husband and wife, the woman may not remarry her first husband, since she had been betrothed by another man in the interim. If they do remarry, in violation of Torah law, then the Bet Din (Rabbinical Court) forces them to divorce. There are certain situations of forbidden marriage where Halacha authorizes the Bet Din to use coercive measures to force the couple of get divorced. The case of "Mahazir Gerushato" where a divorced couple remarries after the woman had been married to somebody else in the interim is an example of such a situation where Bet Din intervenes to force the couple to divorce.
However, if the woman did not marry another man in the interim, then it is permissible and in fact considered a Misva for the husband to remarry her.
If a divorced woman does not marry somebody else, but engages in intimate relations with somebody else without marrying him ("Zenut"), then, ironically enough, she is allowed to remarry her first husband. Even though she had been living with another man, she may remarry her first husband since she was not Halachically married in the interim. Furthermore, if a man and woman lived together without marrying, and then the woman marries another man, she may marry the first man after she and the second man get divorced. The prohibition of "Mahazir Gerushato" applies only if a man and woman were Halachically married and then divorced, and the woman then Halachically marries another man. But if a man and woman lived together without marrying, they may get married even after the woman had married and divorced another man.
Needless to say, these Halachot are quite intricate and complex, and thus in all situations a competent Halachic scholar must be consulted for guidance.
Summary: If a man and woman were married and then got divorced, it is permissible and even a Misva for them to marry each other again, unless the woman was Halachically married to another man in the interim (even if the marriage was never consummated), in which case it is forbidden for her to remarry her first husband.
