The Hijab of a muslim woman in front of mahrams and non mahrams

 A woman must also cover herself fully in front of non-mahram men and stay away from unnecessary mixing with them, in accordance with the Qur'an. Allah says:

"And stay in your homes, and do not display yourselves as was the display of the former times of ignorance." (Qur'an 33:33)

And Allah also says:

"O Prophet, tell your wives, your daughters, and the women of the believers to draw their jalabib (the plural of jilbab, an outer garment) over themselves. That is more suitable so that they will be recognized and not be harmed." (Qur'an 33:59)

 Note: Jalabib is the plural of jilbab, which is an outer garment with full sleeves that covers the body, including the fingernails, and can also cover the face if it is drawn down from the head.

This verse was revealed regarding the best women of this Ummah, who were far removed from immorality even before wearing such garments. Therefore, if they were commanded to observe this level of modesty, then we are even more in need of implementing it in our lives. However, within the privacy of their homes and before those whom Allah has exempted, women are not required to wear the jilbab.

Allah also says in Qur'an 24:31, listing the mahrams before whom a believing woman may display her adornment:

"And tell the believing women to Reduce their gaze and guard their chastity, and not expose their adornment except that which [necessarily] appears thereof, and to draw their khimar (head covering) over their bosoms, and not expose their adornment except to their husbands, their fathers, their husbands' fathers, their sons, their husbands' sons, their brothers, their brothers' sons, their sisters' sons, their women, those whom their right hands possess, male attendants having no physical desire, or children who are not yet aware of the private aspects of women. And let them not stamp their feet to make known what they conceal of their adornment. And turn to Allah in repentance, all of you, O believers, that you might succeed." (Qur'an 24:31)

This verse distinguishes between interactions with mahrams and non-mahrams. 

This Qur'anic verse exempts a woman from going to extremes in front of non-mahrams by allowing her to reveal her face and the parts of her adornment that normally appear, while still maintaining modesty, such as wearing the khimar(hardcover)that covers her bosom but not the face. It then again goes to the extreme for non mahrams by instructing her not even to let the sound of her anklets be heard by non-mahrams while she is in their presence deliberately.