The Portrait of Virgin Mary in the Quran
Islamic narrations have always regarded Mary as one of the four greatest women in the world.  She is also the most frequently mentioned woman in the Quran.
Although the story of Mary is referred to in different chapters of the Quran, chapter 19 is named after her and recounts the story of the miraculous birth of Jesus. And she is always refered to, both in the Quran and Islamic narrations, as a most chaste and pure woman.
But we need to collect Quranic verses from different chapters and put them together in order to make out a complete image of Virgin Mary.
The Worshiping Little Lady
The story begins when Mary's mother is pregnant with her, and she decides to assign her child to working in the temple:
When the wife of Imran said, ‘My Lord, I dedicate to You in consecration what is in my belly. Accept it from me; indeed You are the All-hearing, the All-knowing.’ When she bore her, she said, ‘My Lord, I have borne a female [child]’—and Allah knew better what she had borne—‘and the male [child I expected] was no match for the female [child I have borne], and I have named her Mary, and I commend her and her offspring to Your care against [the evil of] the outcast Satan.’ Thereupon her Lord accepted her with a gracious acceptance, and made her grow up in a worthy fashion... [Quran, 3:35-37]
The priests of the temple disagree on the care taker of the girl. They cast lots, and Zechariah is the lucky one!
... and He charged Zechariah with her care. [Quran, 3:37]
Then we move forward to later years. Mary has turned into a most beautiful girl with such an extraordinary faith and worship which even surprises Prophet Zechariah:
Whenever Zechariah visited her in the sanctuary, he would find provisions with her. He said, ‘O Mary, from where does this come for you?’ She said, ‘It comes from Allah. Allah provides whomever He wishes without any reckoning.’ [Quran, 3:37]
Little by little, Mary hears strange news. To her surprise, she hears that she is chosen and is going to bear a child!
And when the angels said, ‘O Mary, Allah has chosen you and purified you, and He has chosen you above the world’s women. O Mary, be obedient to your Lord and prostrate and bow down with those who bow [in worship].’ ... When the angels said, ‘O Mary, Allah gives you the good news of a Word from Him whose name is Messiah, Jesus son of Mary, distinguished in the world and the Hereafter and one of those brought near [to Allah]. He will speak to the people in the cradle and in adulthood, and will be one of the righteous...’
The news surprises Mary! How could a single woman have a baby? This has never happened before. It's impossible! So, Mary asks her Lord about it:
... She said, ‘My Lord, how shall I have a child seeing that no human has ever touched me?’ He said, ‘So it is that Allah creates whatever He wishes. When He decides on a matter He just says to it ‘‘Be!’’ and it is. [Quran, 3:42-47]
The Pregnant Virgin
Finally, it is time for the prophecies to come true:
And mention in the Book Mary, when she withdrew from her family to an easterly place. Thus did she seclude herself from them, whereupon We sent to her Our Spirit and he became incarnate for her as a well-proportioned human...
The Holy Spirit is, based on the Islamic viewpoint, a creature of God, who is sent by Him to accompany the prophets (and His faithful servants) and show them the best way. Seeing the Holy Spirit, Mary is scared.
... She said, ‘I seek the protection of the All-beneficent from you, should you be Godwary!’ He said, ‘I am only a messenger of your Lord that I may give you a pure son.’ She said, ‘How shall I have a child seeing that no human being has ever touched me, nor have I been unchaste?’ He said, ‘So shall it be. Your Lord says, ‘‘It is simple for Me.’’ And so that We may make him [Jesus] a sign for mankind and a mercy from Us, and it is a matter [already] decided.’ Thus she conceived him, then withdrew with him to a distant place... [Quran, 19:16-22]
The Fruit of Chastity
Mary was chosen from among all girls and women in the world to give birth to Jesus. But this picking was not without reason. This verse states that Mary was chosen since she was the most chaste and pure woman of her time. So, the troubles of protecting one's uprightness and chastity is not without great rewards:
And Mary, daughter of Imran, who guarded the chastity of her womb, so We breathed into it of Our spirit. She confirmed the words of her Lord and His Books, and she was one of the obedient. [Quran, 66:12]
It is narrated from Ja'far ibn Muhammad al-Sadiq that a pretty woman whose looks has put her into sins is brought on the Day of Judgment. She says; "My Lord, You made me pretty, and therefore, I did what I did." So, Mary is brought forth, and it is said to the sinful woman; "Are you prettier than her?! We made her [the most] pretty, but she did not commit any wrongdoing." 
Son of Mary
Renouncing the Christian belief in Jesus divinity and his being "Son of God," the Quran always refers to Jesus as a Word and Spirit from God and "son of Mary:"
O People of the Book! Do not exceed the bounds in your religion, and do not attribute anything to Allah except the truth. The Messiah, Jesus son of Mary, was only an apostle of Allah, and His Word that He cast toward Mary and a spirit from Him... [Quran, 4:171]