Firstly: 

The mahr is a right that the wife has over her husband, whether it is in the form of cash or gold or furniture, as is done in some countries. 

For more information please see the answer to question number 2378. 

Secondly: 

If the man divorces his wife then wants to take her back, one of two scenarios must apply: 

(i)Where he takes her back after the ‘iddah has ended. In this case, there has to be a new marriage contract and a new mahr, with a guardian and two witnesses. In this case the consent of the wife and her guardian is essential.

(ii)Where he takes her back during the ‘iddah. In this case there is no new marriage contract or mahr, and the consent of the wife or her guardian is not required.

For more information please see the answer to question no. 75027. 

Thirdly:  

If a man divorces his wife and he owes her any part of the delayed mahr, or the mahr was in the form of furniture, then he wants to take her back on condition that she give up that mahr, this taking back is valid if the woman agrees to that, because it is her right and she has waived it. The Prophet (blessings and peace of Allah be upon him) said: “The Muslims are bound by their conditions.” Narrated by Abu Dawood (3594) and classed as saheeh by al-Albaani in Saheeh Abi Dawood. 

The woman may decide that staying with her husband is better for her than divorce, so she gives up some of her rights to the husband so that he will keep her and not divorce her. Concerning this, Allah revealed the words (interpretation of the meaning): “And if a woman fears cruelty or desertion on her husband’s part, there is no sin on them both if they make terms of peace between themselves; and making peace is better” [al-Nisa’ 4:128]. ‘Aa’ishah (may Allah be pleased with her) said: This refers to a woman who is married to a man who no longer wants to be intimate with her, so he wants to divorce her and marry someone else, and she says to him: Keep me and do not divorce me, then marry someone else, and you are free of the obligation to spend on me or allocate some of your time to me. This is what Allah says (interpretation of the meaning): “there is no sin on them both if they make terms of peace between themselves; and making peace is better.”

Narrated by al-Bukhaari (5206). 

And Allah knows best.