Can the Wife Go out without Husband’s Permission in Islam?
Question

I want to ask to what degree the husband’s duties towards his wife’s family extend. My question is because I am suffering a serious problem with my husband, because he treated my mother very badly when she came to visit us (because of an argument that took place between my mother-in-law and my mother); in the end my husband virtually threw my mother out. As a result of that, I had to leave the house with my mother, against my husband’s wishes as he wanted me to stay with him. Please note that I used to live in another country and I travelled with my mother to our country. My husband treats me very well, but I got angry when he treated her in this manner. He regretted it the next day, but she will not forgive him. Is what I did correct, or did I fail to obey my husband as enjoined by Allah, may He be exalted and glorified?
Praise be to Allah.Upholding family ties and treating in-laws kindly
The husband should uphold ties with his wife’s family and treat them well. This is part of treating his wife kindly, because doing that makes her happy and earns him respect in her eyes, and increases the love and affection between them. 
Allah says (interpretation of the meaning): “…and live with them honourably.” [An-Nisa’ 4:19]
Ibn Kathir said:  
“I.e., speak kindly to them, treat them well and pay attention to your deeds and your appearance as much as you can; as you would like her to do for you, do the same for her. Allah says (interpretation of the meaning): 
“And they (women) have rights (over their husbands as regards living expenses) similar (to those of their husbands) over them (as regards obedience and respect) to what is reasonable.” [Al-Baqarah 2:228]
The Messenger of Allah (peace and blessings of Allah be upon him) said: “The best of you is the one who is best towards his family; and I am the best of you towards my family.” (Classed as authentic by Al-Albani in As-Silsilah As-Sahihah, 285." (Tafsir Ibn Kathir, 1/477 )
The role of apology and forgiveness in marriage
With regard to your husband throwing your mother out of his house, he has apologised for that, and if a person apologises, the apology should be accepted and his mistake should be overlooked. 
The married woman should remember that obedience to her husband takes precedence over obedience to her parents. A man should not give precedence to anyone over his mother with regard to kind treatment, and a woman should not give precedence to anyone over her husband with regard to obedience. That is because of the greatness of the rights that he has over her. Part of the greatness of men’s rights over women is that Shari`ah almost commanded women to prostrate to men, were it not for the fact that it is not permissible for anyone to prostrate to any human being. 
The husband has no right to prevent his wife’s family from visiting their daughter , unless he fears that they may cause some mischief to her or encourage her to be willfully defiant towards him (Nashiz). In that case, he may prevent visits. 
Wife going out without the husband’s permission in Islam 
You made two mistakes and went against Shari`ah by doing them. The first mistake was going out of the house without your husband’s permission , and the second was travelling without a Mahram. 
Going out of the house without the husband’s permission is a prohibited action; Allah has even forbidden women who are revocably divorced (first or second divorce) from going out of their houses, so how about women who are not in that position? Allah says (interpretation of the meaning): 
“O Prophet! When you divorce women, divorce them at their `Iddah (prescribed periods) and count (accurately) their `Iddah (periods). And fear Allah your Lord (O Muslims). And turn them not out of their (husband’s) homes nor shall they (themselves) leave, except in case they are guilty of some open illegal sexual intercourse. And those are the set limits of Allah. And whosoever transgresses the set limits of Allah, then indeed he has wronged himself.” [At-Talaq 65:1]
Shaykh Al-Islam Ibn Taymiyah said: 
“Zayd ibn Thabit said: The husband is the master according to the Book of Allah, and he recited the verse in which Allah says (interpretation of the meaning): 
“They both found her lord [sayyid] (i.e. her husband) at the door.” [Yusuf 12:25]
`Umar ibn Al-Khattab said: “Marriage is slavery, so be careful with regard to whom you give your daughter for enslavement.” In At-Tirmidhi and elsewhere it is narrated that the Prophet (peace and blessings of Allah be upon him) said: “I urge you to treat women well, for they are prisoners with you.”  
So a woman is like a slave or prisoner of her husband, and she cannot go out of his house except with his permission, whether her father, her mother or anyone else tells her to do that, according to the consensus of the imams." (Al-Fatawa Al-Kubra, 3/148)
Ibn Muflih Al-Hanbali said: 
“It is prohibited for a woman to go out of her husband’s house without his permission, except in cases of necessity, or Shar`i obligations." (Al-Adab Ash-Shar`iyyah, 3/375)
With regard to a woman travelling without a mahram , this is prohibited. This is stated in authentic Hadiths from the Prophet (peace and blessings of Allah be upon him). 
An-Nawawi said: 
“Everything that is called travelling, it is forbidden for a woman to do without her husband or a mahram, whether it is three days, two days or one day, or Barid (a distance equivalent to approximately twenty kilometres) or anything else, because of the Hadith of Ibn `Abbas, according to which the Prophet (peace and blessings of Allah be upon him) said: “No woman should travel without a Mahram.” This includes everything that is called travel. And Allah knows best.” (Sharh Muslim, 9/103) 
And Allah knows best.