Zabur (Arabic: زبور) is, according to Islam, the holy book of Dawud (David), one of the holy books (or religious text) revealed by God before the Qur'an, alongside others such as the Tawrat (Torah) of Musa (Moses) and the Injil (Gospel) of Isa (Jesus).
Some scholars equate the Zabur with the biblical book of Psalms. The term zabur could be either the Arabic equivalent of the Hebrew zimrah, meaning "song, music." which, along with zamir ("song"; "nightingale") and mizmor ("psalm"), is a derivative of zamar, meaning "singer, praiser, music maker or, of sipur (a story; an account) which could have also been heard pronounced as: sapur (with the linguistically-known phenomenon of interchangeable minimal pair sound of the voiceless & voiced letters: p & b, respectively, between Hebrew & Arabic)."
The Zabur of Dawud (David) is referred to in the Qur'an as one of God's books revealed to four selected messengers. The Zabur is preceded by the Taurat (Torah) given to Moses and followed by the Injeel (Gospel) given to Jesus and finally the Qur'an given to the Islamic prophet, Muhammad.
According to Islam, it has been there even since the time of Adem (Adam, who is considered the first human) and not beginning in the 7th century. The Aqeedah (belief system) of Islam is believed to be exactly the same in every single one of the four scriptures and several scrolls and is thought exactly the same by every prophet. Considering this it can be assumed the theological aspects of the Zabur be exactly the same as the Qur'an.
The Sharia or laws of Islam however are known to have changed slightly depending on the prophet of the time, although the Sharia that is revealed in the Qur'an is final till the end of time as there will be no prophet after Muhammad, and no book after the Qur'an. Hence the laws of the Zabur need not be exactly the same as what is Islam of today (as in the Sharia of the Qur'an).
Mention of Zabur in the Qur'an
In the Qur'an, the Zabur is mentioned by name only three times. The Qur'an itself says nothing about the Zabur specifically, except that it was revealed to David, king of Israel and that in Zabur is written "My servants the righteous, shall inherit the earth".
We have sent thee inspiration, as We sent it to Noah and the Messengers after him: we sent inspiration to Abraham, Isma'il, Ishaq, Younus and the Tribes, to Isa, Job, Yunus, Aaron, and Sulayman, and to Dawood We gave the Psalms.
And it is your Lord that knoweth best all beings that are in the heavens and on earth: We did bestow on some prophets more (and other) gifts than on others: and We gave to David (the gift of) the Psalms.
Before this We wrote in the Psalms, after the Message (given to Moses): My servants the righteous, shall inherit the earth."
Parallel of ayah 21:105 with Psalms
The last reference is of interest because of the quotation from Psalm 37 verse 29 which says, "The righteous shall inherit the land, and dwell therein for ever," (as translated in the King James Version of the Bible).
According to Ahrens (1930) the last reference is quoted from Psalms. He says that the verse in the Qur'an reads "We have written in the Zabur after the reminder that My righteous servants shall inherit the earth." His conclusion is that this verse represents a close and rare linguistic parallel with the Hebrew Bible and more pointedly, with Psalm 37 ascribed specifically to David (see verses 9, 11, 29 which refer to the meek, the righteous or “those who wait upon the Lord” as they who shall inherit the earth).
Many Muslim scholars think that it also has reference to Exodus 32:13, which reads "Remember Abraham, Isaac, and Israel, thy servants, to whom thou swearest by thine own self, and saidst unto them, I will multiply your seed as the stars of heaven, and all this land that I have spoken of will I give unto your seed, and they shall inherit it for ever."
Zabur in Sunni Hadith
One hadith, considered valid by Muhammad al-Bukhari, says:
Narrated Abu Huraira: The Prophet said, "The reciting of the Zabur (i.e. Psalms) was made easy for David. He used to order that his riding animals be saddled, and would finish reciting the Zabur before they were saddled. And he would never eat except from the earnings of his manual work."
Zabur and Ketuvim
Christian apologist Karl Gottlieb Pfander suggested that the Qur'an's reference to Zabur actually refers to the third division of the Hebrew Scriptures, known as the Writings or Ketuvim, a broader grouping of Jewish holy books encompassing the Psalms and other collections of Hebrew literature and poetry. 
- Theological Wordbook of the Old Testament, vol. 1, pg. 245.
- Psalms 37:29
- Quran 21:105 (Translated by Yusuf Ali)
- Quran 4:163 (Translated by Yusuf Ali)
- See also Ibrāhīm, Ismā'īl, Ishaq, Jakub, Ayyub, Yunus, Harun and Sulayman
- Quran 17:55 (Translated by Yusuf Ali)
- K. Ahrens, Christliches im Qoran, in ZDMG , lxxxiv (1930), 29
- Psalms 37:9
- Psalms 37:11
- Exodus 32:13
- C. G. Pfander, The Balance of Truth, pg. 51