Three Quranic passages carry the discussion, and it helps to see what each one contains.
Sura 24, verse 31 instructs believing women to lower their gaze, guard their modesty, not display their adornment except what ordinarily appears, and draw their khimar over their bosoms, then lists the relatives before whom the restriction does not apply. The khimar is a head covering already in use; the instruction concerns where it is to be drawn, namely over the chest rather than thrown back. Classical exegesis reads the head covering as assumed by the verse and the command as extending it. A body of modern scholarship reads the same fact differently, arguing that if the head covering were the point the verse would have commanded it rather than presupposing a garment and legislating its arrangement.
Sura 33, verse 59 tells the Prophet's wives, daughters and the believing women to draw their jilbab about them, so that they may be recognised and not molested, which attaches a stated purpose to the instruction and has generated a long argument about whether the ruling travels with the purpose.
Sura 33, verse 53 uses the word hijab itself, and it means a curtain: visitors to the Prophet's house are to ask for what they need from behind it. The word that names the garment in modern usage does not, in the Quran, refer to a garment at all.
The classical schools converged on the position that a woman covers all but the face and hands before unrelated men, with disagreements at the margins. Contemporary scholars including Amina Wadud and Asma Barlas have argued for readings on which the covering is culturally located rather than universally binding. Both positions are held by people arguing from the same texts, which is the honest description.