بسم الله الرحمن الرحيم
Rulings of Hijāb (from Tafseer Al-Qur'ān)

The following is a summary commentary of soorah an-Noor, āyah 31 from Tafseer Ibn Katheer.
 
This āyah is a command from Allāh to the believing women, and jealousy on His part over the wives of His believing servants.  It is also to distinguish the believing women from the women of the Jahiliyyah and the deeds of the pagan women.  Allāh subhānahu wa ta a'lā revealed:

﴿وَقُل لِّلْمُؤْمِنَـتِ يَغْضُضْنَ مِنْ أَبْصَـرِهِنَّ﴾

"And tell the believing women to lower their gaze" meaning, from that which Allāh has forbidden them to look at, apart from their husbands.
[Some] scholars said that it is permissible for women to look at non-Mahram men without desire, as it was recorded in the Sahih that the Messenger of Allāh was watching the Ethiopians playing with spears in the Masjid on the day of `Eed, and `A'ishah the Mother of the believers was watching them from behind him and he was concealing her from them, until she got bored and went away.

﴿وَيَحْفَظْنَ فُرُوجَهُنَّ﴾

"and protect their private parts". Sa`id bin Jubayr said: "From immoral actions.'' Abu Al-`Aliyah said: "Every āyah of the Qur'ān in which protecting the private parts is mentioned means protecting them from Zina, except for this āyah which means protecting them from being seen by anybody.''

﴿وَلاَ يُبْدِينَ زِينَتَهُنَّ إِلاَّ مَا ظَهَرَ مِنْهَا﴾

"and not to show off their adornment except that which is apparent" means they should not show anything of their adornment to non-Mahram men except for whatever it is impossible to hide.  Ibn Mas`ud said: "Such as clothes and outer garments,'' Meaning what the Arab women used to wear of the veil which covered their clothes and whatever showed from underneath the outer garment.  There is no blame on her for this, because this is something that she cannot conceal.  Similar to that is what appears of her lower garment and what she cannot conceal.  Al-Hasan, Ibn Sirin, Abu Al-Jawza', Ibrahim An-Nakha`i and others also had the same view as Ibn Mas`ud.

﴿وَلْيَضْرِبْنَ بِخُمُرِهِنَّ عَلَى جُيُوبِهِنَّ﴾

"and to draw their veils all over their Juyub" means that they should wear the outer garment in such a way as to cover their chests and ribs, so that they will be different from the women of the Jahiliyyah, who did not do that but would pass in front of men with their chests completely uncovered, and with their necks, forelocks, hair and earrings uncovered.  So Allāh commanded the believing women to cover themselves, as He says:

﴿يأَيُّهَا النَّبِىُّ قُل لاًّزْوَجِكَ وَبَنَـتِكَ وَنِسَآءِ الْمُؤْمِنِينَ يُدْنِينَ عَلَيْهِنَّ مِن جَلَـبِيبِهِنَّ ذلِكَ أَدْنَى أَن يُعْرَفْنَ فَلاَ يُؤْذَيْنَ﴾

"O Prophet!  Tell your wives and your daughters and the women of the believers to draw their cloaks all over their bodies. That will be better, that they should be known, so as not to be annoyed" [33:59]
 
And in this noble āyah Allāh subhānahu wa ta'āla said:

﴿وَلْيَضْرِبْنَ بِخُمُرِهِنَّ عَلَى جُيُوبِهِنَّ﴾

"and to draw their (Khumur) veils all over their Juyub".  Khumur (veils) is the plural of Khimar, which means something that covers, and is what is used to cover the head. This is what is known among the people as a veil.  Sa`id bin Jubayr said: "and to draw" means to pull it around and tie it securely.  "their veils all over their Juyub" means, over their necks and chests so that nothing can be seen of them.  Al-Bukhari recorded that `A'ishah, may Allāh be pleased with her, said: "May Allāh have mercy on the women of the early emigrants.  When Allāh revealed the āyah, they tore their aprons and Akhtamar themselves with them.''  He also narrated from Safiyyah bint Shaybah that `A'ishah, may Allāh be pleased with her, used to say: "When this āyah was revealed, they took their Izars (waistsheets) and tore them at the edges, and Akhtamar themselves with them.''

﴿وَلاَ يُبْدِينَ زِينَتَهُنَّ إِلاَّ لِبُعُولَتِهِنَّ أَوْ ءَابَآئِهِنَّ أَوْ ءَابَآءِ بُعُولَتِهِنَّ أَوْ أَبْنَآئِهِنَّ أَوْ أَبْنَآءِ بُعُولَتِهِنَّ أَوْ إِخْوَانِهِنَّ أَوْ بَنِى إِخْوَانِهِنَّ أَوْ بَنِى أَخَوَتِهِنَّ﴾

"and not to reveal their adornment except to their husbands, or their fathers, or their husband's fathers, or their sons, or their husband's sons, or their brothers or their brother's sons, or their sister's sons,"  All of these are a woman's close relatives whom she can never marry (Mahram) and it is permissible for her to show her adornments to them, but without making a wanton display of herself.

﴿أَوْ نِسَآئِهِنَّ﴾

"or their women," this means that she may also wear her adornment in front of other Muslim women, but not in front of the women of Ahl Adh-Dhimmah (Jewish and Christian women), lest they describe her to their husbands.  This is prohibited for all women, but more so in the case of the women of Ahl Adh-Dhimmah, because there is nothing to prevent them from doing that, but Muslim women know that it is unlawful and so, would be deterred from doing it. The Messenger of Allāh said:

«لَا تُبَاشِرِ الْمَرْأَةُ الْمَرْأَةَ فَتَنْعَتَهَا لِزَوْجِهَا كَأَنَّهُ يَنْظُرُ إِلَيْهَا»

"No woman should describe another woman to her husband so that it is as if he is looking at her."  It was recorded in the Two Sahihs from Ibn Mas`ud.

﴿أَوْ مَا مَلَكَتْ أَيْمَـنُهُنَّ﴾

"or their right hand possessions."  Ibn Jarir said, "This means from among the women of the idolators. It is permissible for a Muslim woman to reveal her adornment before such a woman, even if she is an idolatress, because she is her slave-girl.'' This was also the view of Sa`id bin Al-Musayyib. Allāh says;

﴿أَوِ التَّـبِعِينَ غَيْرِ أُوْلِى الإِرْبَةِ مِنَ الرِّجَالِ﴾

"Tabi`in among men who do not have desire," such as hired servants and followers who are not at the same level as the woman and are feeble-minded and have no interest in or desire for women.  Ibn `Abbas said, "This is the kind of person who has no desire.'' 

﴿أَوِ الطِّفْلِ الَّذِينَ لَمْ يَظْهَرُواْ عَلَى عَوْرَتِ النِّسَآءِ﴾

"or children who are not aware of the nakedness of women."  Because they are so young they do not understand anything about women or their `Awrah or their soft speech or their enticing ways of walking and moving.  If a child is small and does not understand that, there is nothing wrong with him entering upon women, but if he is an adolescent or approaching adolescence, so that he knows and understands these things, and can make a distinction between who is beautiful and who is not, then he should not enter upon women.  It was recorded in the Two Sahihs that the Messenger of Allāh said:

«إِيَّاكُمْ وَالدُّخُولَ عَلَى النِّسَاءِ»

"Avoid entering upon women." It was said, "O Messenger of Allāh, what do you think about the male in-laws'' He said:

«الْحَمْوُ: الْمَوْتُ»

"The male in-law is death." ("Al-Hamoo" refers to the male in-laws that the woman would be permitted to marry if she were divorced from her husband, e.g. brother-in-law.)
 
 
References
Source: Tafseer Ibn Katheer (Abridged English Translation), Volume 7.
 
For further information click on link: Why Hijāb.