Women who are or have been married are required to cover their hair. An
unmarried virgin woman is not. There is some disagreement on the status
of an unmarried virgin (and vice versa with kiddushin but not nisu'in,
not really relevant today).
The Talmud in Kesuvos 72a states that the source for this prohibition is
from BaMidbar (Numbers) 5:18 which deals with the laws of a sotah -
suspected adulteress - and states, "The priest shall stand the woman
before God and uncover her hair..." Rash'i provides two explanations for
the Talmud's conclusion, one, that from the fact that this was done as a
punishment midah kneged midah (measure for measure) because she had
exposed her hair to her paramour we see that this is prohibited and, two,
from the fact that we expose her hair we see that under normal conditions
a Jewish woman's hair should be covered.
For a more in depth discussion of the prohibition against a woman
exposing her hair and the ethical/moral issues involved I suggest you
read the Chofetz Chaim's book Geder Olam which is devoted to this
subject. I believe it has been translated into English but I'm not sure
how available the translation is.
Eliezer C. Abrahamson
Talmud Torah - Basic Jewish Education