Short answer
Islam honours Mary above all other women. She is the only woman the Qur'an names, and it names her 34 times. That is more often than the whole New Testament names her. A chapter of the Qur'an is called by her name. The angels told her that God (﷾) had chosen her, purified her, and chosen her above the women of all nations. They brought her news of a son. She asked how, since no man had touched her, and was told that God (﷾) creates what He wills. When her people accused her, the baby Jesus spoke from the cradle to defend her. The Prophet (ﷺ) called her the best of the women of her time. Muslims honour her but never worship her, and no Muslim may speak of her with anything but reverence.
Full explanation
Chosen above the women of all nations
The Qur'an tells Mary's story from before her birth. Her mother gave her to God (﷾), named her Mary and asked God (﷾) to protect her and her child from Satan. God (﷾) accepted her, and she grew up in purity, in the care of Zakariya, a prophet. Later the angels told her that God (﷾) had chosen her, purified her, and chosen her above the women of all nations. They told her to worship her Lord.
“When she was delivered, she said: "O my Lord! Behold! I am delivered of a female child!"- and Allah knew best what she brought forth- "And no wise is the male Like the female. I have named her Mary, and I commend her and her offspring to Thy protection from the Evil One, the Rejected."”
“Right graciously did her Lord accept her: He made her grow in purity and beauty: To the care of Zakariya was she assigned. Every time that he entered (Her) chamber to see her, He found her supplied with sustenance. He said: "O Mary! Whence (comes) this to you?" She said: "From Allah: for Allah Provides sustenance to whom He pleases without measure."”
“Behold! the angels said: "O Mary! Allah hath chosen thee and purified thee- chosen thee above the women of all nations.”
“"O Mary! worship Thy Lord devoutly: Prostrate thyself, and bow down (in prayer) with those who bow down."”
The virgin birth
The Qur'an teaches the virgin birth plainly. The angels gave Mary news of a Word from God (﷾) whose name would be the Messiah, Jesus, son of Mary. She asked how she could have a son when no man had touched her. She was told that God (﷾) creates what He wills. He decrees a thing and says to it "Be", and it is. The chapter called Mary tells the same story. It then tells how God (﷾) comforted her in the pains of childbirth under a palm tree.
“Behold! the angels said: "O Mary! Allah giveth thee glad tidings of a Word from Him: his name will be Christ Jesus, the son of Mary, held in honour in this world and the Hereafter and of (the company of) those nearest to Allah;”
“She said: "O my Lord! How shall I have a son when no man hath touched me?" He said: "Even so: Allah createth what He willeth: When He hath decreed a plan, He but saith to it, 'Be,' and it is!”
“Relate in the Book (the story of) Mary, when she withdrew from her family to a place in the East. She placed a screen (to screen herself) from them; then We sent her our angel, and he appeared before her as a man in all respects. She said: "I seek refuge from thee to (Allah) Most Gracious: (come not near) if thou dost fear Allah." He said: "Nay, I am only a messenger from thy Lord, (to announce) to thee the gift of a holy son. She said: "How shall I have a son, seeing that no man has touched me, and I am not unchaste?" He said: "So (it will be): Thy Lord saith, 'that is easy for Me: and (We wish) to appoint him as a Sign unto men and a Mercy from Us': It is a matter (so) decreed."”
“So she conceived him, and she retired with him to a remote place. And the pains of childbirth drove her to the trunk of a palm-tree: She cried (in her anguish): "Ah! would that I had died before this! would that I had been a thing forgotten and out of sight!" But (a voice) cried to her from beneath the (palm-tree): "Grieve not! for thy Lord hath provided a rivulet beneath thee; "And shake towards thyself the trunk of the palm-tree: It will let fall fresh ripe dates upon thee. "So eat and drink and cool (thine) eye. And if thou dost see any man, say, 'I have vowed a fast to (Allah) Most Gracious, and this day will I enter into not talk with any human being'"”
Jesus defended his mother from the cradle
When Mary brought the baby to her people, they accused her of a shameful thing. She pointed to the child. Jesus spoke from the cradle. He called himself a servant of Allah (﷾) and a prophet, and said that God (﷾) had made him kind to his mother. The Qur'an also condemns those who spoke a grave false charge against Mary. Her people called her "sister of Aaron". Christians in Najran, a town in Arabia, put this question to a Companion, al-Mughira ibn Shu'ba. The Prophet (ﷺ) explained to him that people were named after the prophets and righteous people before them. The page "Why does the Qur'an call Mary the 'sister of Aaron'?" answers this in full.
“At length she brought the (babe) to her people, carrying him (in her arms). They said: "O Mary! truly an amazing thing hast thou brought! "O sister of Aaron! Thy father was not a man of evil, nor thy mother a woman unchaste!" But she pointed to the babe. They said: "How can we talk to one who is a child in the cradle?" He said: "I am indeed a servant of Allah: He hath given me revelation and made me a prophet; "And He hath made me blessed wheresoever I be, and hath enjoined on me Prayer and Charity as long as I live; "(He) hath made me kind to my mother, and not overbearing or miserable; "So peace is on me the day I was born, the day that I die, and the day that I shall be raised up to life (again)"!”
“That they rejected Faith; that they uttered against Mary a grave false charge;”
Al-Mughira ibn Shu'ba says the Christians of Najran asked him how the Qur'an could call Mary "sister of Aaron" when Moses lived long before Jesus. He put the question to the Prophet (ﷺ), who answered that people used to name their children after the prophets and righteous people who had gone before them.
A sign for all peoples
The Qur'an holds Mary up as an example to all believers. She guarded her chastity, believed the words of her Lord and was one of the devout. God (﷾) made her and her son a sign for all peoples. The Prophet (ﷺ) said Mary was the best of the women of her time. He also said that Satan did not touch her or her son at birth, as her mother had prayed.
“And Mary the daughter of 'Imran, who guarded her chastity; and We breathed into (her body) of Our spirit; and she testified to the truth of the words of her Lord and of His Revelations, and was one of the devout (servants).”
“And (remember) her who guarded her chastity: We breathed into her of Our spirit, and We made her and her son a sign for all peoples.”
Ali reports that he heard the Prophet (ﷺ) say that Mary, the daughter of Imran, was the best of the women of her time, and Khadija the best of the women of his own community.
Abu Huraira reports that the Prophet (ﷺ) said: Satan touches every child of Adam at birth, and so the newborn cries out, except Mary and her son. Abu Huraira then recited the prayer of Mary's mother in Qur'an 3:36, asking protection for her and her offspring from Satan.
Named more often in the Qur'an than in the New Testament
The Arabic Qur'an names Mary 34 times. Most of these are in the title it gives Jesus, 'son of Mary'. The New Testament uses that title only once (Mark 6:3). The New Testament names the mother of Jesus 19 times in the King James Version, almost all of them in Matthew and Luke. The Gospel of John never gives her name, and neither do Paul's letters.
“Relate in the Book (the story of) Mary, when she withdrew from her family to a place in the East. She placed a screen (to screen herself) from them; then We sent her our angel, and he appeared before her as a man in all respects. She said: "I seek refuge from thee to (Allah) Most Gracious: (come not near) if thou dost fear Allah." He said: "Nay, I am only a messenger from thy Lord, (to announce) to thee the gift of a holy son. She said: "How shall I have a son, seeing that no man has touched me, and I am not unchaste?" He said: "So (it will be): Thy Lord saith, 'that is easy for Me: and (We wish) to appoint him as a Sign unto men and a Mercy from Us': It is a matter (so) decreed."”
“And Mary the daughter of 'Imran, who guarded her chastity; and We breathed into (her body) of Our spirit; and she testified to the truth of the words of her Lord and of His Revelations, and was one of the devout (servants).”
“Then said Mary unto the angel, How shall this be, seeing I know not a man? And the angel answered and said unto her, The Holy Ghost shall come upon thee, and the power of the Highest shall overshadow thee: therefore also that holy thing which shall be born of thee shall be called the Son of God.”
How Christians honour Mary
Christians also honour Mary as the virgin mother of Jesus. The Council of Ephesus, a church council held in 431, confirmed the title Mother of God. Many Christians use it, reading in that way the greeting of Elizabeth, Mary's relative, to "the mother of my Lord". In Luke, Mary says that all generations will call her blessed.
“And whence is this to me, that the mother of my Lord should come to me?”
“For he hath regarded the low estate of his handmaiden: for, behold, from henceforth all generations shall call me blessed.”
Honoured, never worshipped
Muslims call Mary blessed in every generation, as she said they would. The Qur'an also draws a firm line. Mary was a woman of truth who, like her son, needed to eat. On the Day of Judgement, Jesus will deny ever telling anyone to worship him and his mother as gods.
“For he hath regarded the low estate of his handmaiden: for, behold, from henceforth all generations shall call me blessed.”
“Christ the son of Mary was no more than a messenger; many were the messengers that passed away before him. His mother was a woman of truth. They had both to eat their (daily) food. See how Allah doth make His signs clear to them; yet see in what ways they are deluded away from the truth!”
“And behold! Allah will say: "O Jesus the son of Mary! Didst thou say unto men, worship me and my mother as gods in derogation of Allah'?" He will say: "Glory to Thee! never could I say what I had no right (to say). Had I said such a thing, thou wouldst indeed have known it. Thou knowest what is in my heart, Thou I know not what is in Thine. For Thou knowest in full all that is hidden.”
Perspectives
Christian
Mary is the virgin mother of Jesus, highly favoured by God. Many churches call her the Mother of God, because they hold her son to be God. The Council of Ephesus confirmed that title in 431. Catholic and Orthodox Christians also ask for her prayers.
Luke records that Mary conceived as a virgin by the power of the Holy Spirit.
Rebuttal
Islam agrees: the Qur'an teaches the virgin birth plainly. But a birth without a father makes no one divine, for God (﷾) creates what He wills by saying "Be". And from the cradle Jesus called himself a servant of Allah (﷾).
Elizabeth calls Mary "the mother of my Lord", which many Christians read as calling her the Mother of God.
Rebuttal
The title "Mother of God" rests on the belief that her son is God, not on her honour. From the cradle, the Qur'an's Jesus calls himself a servant of Allah (﷾), and both he and his mother needed to eat. On the Day of Judgement, Jesus will deny telling anyone to worship him and his mother as gods.
The Islamic perspective
Mary is the most honoured woman in the Qur'an: chosen, purified, a virgin mother and a sign for all peoples. The Prophet (ﷺ) called her the best of the women of her time. She is honoured as God's (﷾) devout servant, never worshipped.
The Qur'an says God (﷾) chose Mary, purified her, and chose her above the women of all nations.
The Qur'an teaches that Mary conceived Jesus as a virgin, by God's (﷾) command.
Mary is the only woman named in the Qur'an, and a chapter is called by her name.
The Qur'an names Mary 34 times, more than the New Testament names her.
The Qur'an records Jesus speaking from the cradle to clear his mother's name, and condemns the slander against her.
The Prophet Muhammad (ﷺ) called Mary the best of the women of her time.
Mary ate food like any human being, and Jesus never told anyone to worship her.