The Arabic word “Hijab” literally means veil, screen, curtain or cover. 
The modesty in Qur'an concerns both men's and women's gaze, walk, garments, and genitalia. Hijab is first for men and then for women. Muslims believe that their sole purpose in life is to worship of Allah (Glory be to Him) alone, according to His instructions, as revealed in the Glorious Qur’an, and through the teachings of the Prophet Mohammad (May Allah honour Him and grant Him peace). Hijab is an act of obedience to Allah (Glory be to Him), “It is not for a believer, man or woman, when Allah (Glory be to Him) and His Messenger (May Allah honour Him and Grant Him peace) have decreed a matter that they should have an option in their decision. And whoever disobeys Allah (Glory be to Him) and His Messenger has indeed strayed in a plain error.” Qur’an.Surah Ahzab 33:36. 
The Qur’an mentions that Hijab has been prescribed for the women so that they are recognized as modest women and it will also prevent them from being molestated. Modesty is the symbol of faith and whoever has no modesty (Hijab), has no religion. Hijab does not degrade a woman but uplifts a woman and protects her modesty and chastity. Hijab generate respect for each other and help in creating peaceful society.
Hijab refers to both covering and veil as Allah Himself uses in the Qur’an, And it is not for any human being that Allah should speak to him except by revelation or from behind a partition (veil) Qur’an 42:51
And say to the believing women that they should lower their gaze and guard their modesty; that they should not display their beauty and ornaments except what (must ordinarily) appear thereof; that they should draw their veils over their bosoms and not display their beauty except to their husbands, their fathers, their husband's fathers, their sons, their husbands' sons, their brothers or their brothers' sons, or their sisters' sons, or their women, or the slaves whom their right hands possess, or male servants free of physical needs, or small children who have no sense of the shame of sex; and that they should not strike their feet in order to draw attention to their hidden ornaments. Qur’an.Surah Nur 24:31.
O Prophet, tell your wives and your daughters and the women of the believers to bring down over themselves [part] of their outer garments. That is more suitable that they will be known and not be abused. And ever is Allah Forgiving and Merciful. Qur’an.Surah Ahzab 33:59 The word Jilbab (cloak) means the cover over the veil (yashmak)
And abide in your houses and do not display yourselves as [was] the display of the former times of ignorance. And establish prayer and give zakah and obey Allah and His Messenger. Allah intends only to remove from you the impurity [of sin], O people of the [Prophet's] household, and to purify you with [extensive] purification. Qur’an.Surah Ahzab 33:33 
…And when you ask [his wives] for something, ask them from behind a partition. That is purer for your hearts and their hearts. Qur’an.Surah Ahzab 33:53. 
O Prophet, tell your wives and your daughters and the women of the believers to bring down over themselves [part] of their outer garments. That is more suitable that they will be known and not be abused. Qur’an.Surah Ahzab 33:59. 
It was narrated from Safiyyah bint Shaybah® that ‘Aa’ishah(R) used to say: When these words were revealed – “and to draw their veils all over Juyoobihinna (i.e. their bodies, faces, necks and bosoms)” – they took their izaars (a kind of garment) and tore them from the edges and covered their faces with them. Sahih Al Bukhaari 4481. 
Narrated from ‘Abd-Allaah that the Prophet (peace and blessings of Allaah be upon him) said: “The woman is ‘awrah and when she goes out the Shaytaan gets his hopes up.” al-Tirmidhi, 1173, Al-Albaani said in Saheeh al-Tirmidhi (936): It is saheeh.
Narrated that ‘Aa’ishah (may Allaah be pleased with her) said: “May Allaah have mercy on the women of the Muhaajireen. When Allaah revealed the words (interpretation of the meaning) ‘and to draw their veils all over Juyoobihinna (i.e. their bodies, faces, necks and bosoms)…” [al-Noor 24:31], they tore their aprons and covered their faces with them.” (Narrated by al-Bukhaari, 4480)
Aa’ishah said: “The riders used to pass by us when we were with the Messenger of Allaah (peace and blessings of Allaah be upon him) in ihraam. When they came near, each of us would lower her jilbaab from her head over her face, and when they passed by we would uncover (our faces).” Abu Dawood, 1833; Ahmad, 24067 , Shaykh al-Albaani said in Jilbaab al-Mar’ah al-Muslimah (107): its isnaad is hasan 
In the Qur’an, Allah (Glory be to Him) states: Say to the believing men that they should lower their gaze and guard their modesty: that will make for greater purity for them: And Allah is well acquainted with all that they do. Qur’an.Surah Nur 24:30
It is not permissible to shake hands with a non-Mahram, because the Prophet Muhammad (May Allah honour Him and Grant Him peace) said: "For one of you to be stabbed in the head with an iron needle is better for him than to touch a woman who is not permissible for him.” At-Tabaraani in Al-Mu’jam Al-Kabir (20/213), Ibn Abi Shaybah in Al-Musannaf (4/341)
In the Qur’an, Allah (Glory be to Him) states:
And tell the believing women to reduce [some] of their vision and guard their private parts and not expose their adornment except that which [necessarily] appears thereof and to wrap [a portion of] their headcovers over their chests 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, that which their right hands possess, or those 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.
There are three signs of puberty that males and females have in common:
1 – Nocturnal emissions (wet dreams)
2 – Growth of coarse hairs around the private parts
3 – Reaching the age of fourteen (In Arabs it is less that that)
In the case of females there is a fourth sign, which is:
4 – Menstruation
If one of these signs of puberty appears in a girl, then she is obliged to do all the obligatory duties and to avoid all haraam things. One of the obligatory duties is wearing hijab. 
The reason for this strictness is so that the woman is protected from the lustful gaze of men. She should not attract attention to herself in any way. It is permissible for a man to catch the eye of a woman, however it is haram (unlawful) for a man to look twice as this encourages lustful thoughts.
Islam protects the woman, it is for this reason that Allah (Glory be to Him) gave these laws. In today's society womankind is being exploited, female sexuality is being openly used in advertising, mainly to attract the desires of men and therefore sell the product. Is the woman really free in today's society? The answer is obviously no, the constant bombardment by the media as to how the ideal woman should look and dress testifies to this. Islam liberated woman over 1400 years ago. 
The hijab reflects modesty, purity and respect; it lessens temptation so that more serious sins will be avoided; it protects women from the harm and molestation of evil men; A woman who covers her head is making a statement that she is a member of the Muslim community and that she follows a particular code of moral conduct.
Once a woman begins to wear hijab she completes a large portion of her responsibility for dawah with very little effort. Each time that she goes to the grocery store, the library, to work, to school, or to any other public place, she is spreading the magnificent message of Islam. This is not only because of the outer hijab that she wears, but more importantly, the modesty of her behavior that accompanies it. When a woman refrains from flirting with men, limits physical contact, and is reserved and respectful, people may become curious and want to learn more about this intriguing faith. Dr. Aisha Hamdan, AlJumuah Magazine, Volume 10, Issue 5, Jumaada-A- Ulaa, 1419 A.H
“Allaah commands women to let the jilbaab come down (over their faces) so that they will be known (as respectable women) and not be annoyed or disturbed. This evidence supports the first opinion. ‘Ubaydah al-Salmaani and others stated that the women used to wear the jilbaab coming down from the top of their heads in such a manner that nothing could be seen except their eyes, so that they could see where they were going. It was proven in al-Saheeh that the woman in ihraam is forbidden to wear the niqaab and gloves. This is what proves that the niqaab and gloves were known among women who were not in ihraam. This implies that they covered their faces and hands.” (Majmoo’ al-Fataawa, 15/371-372)
Shaykh Muhammad al-Saalih al-‘Uthaymeen (may Allaah have mercy on him) said:
The hijaab prescribed in sharee’ah means that a woman should cover everything that it is haraam for her to show, i.e., she should cover that which it is obligatory for her to cover, first and foremost of which is the face, because it is the focus of temptation and desire.
A woman is obliged to cover her face in front of anyone who is not her mahram (blood relative to whom marriage is forbidden). From this we learn that the face is the most essential thing to be covered. There is evidence from the Book of Allaah and the Sunnah of His Prophet (peace and blessings of Allaah be upon him) and the views of the Sahaabah and the imams and scholars of Islam, which indicates that women are obliged to cover all of their bodies in front of those who are not their mahrams. Fataawa al-Mar’ah al-Muslimah, 1/ 391, 392 
Hijab by Abdulaziz Addwesh,
  http://islamqa.info/en/ref/21536