Where does the tradition of nun's veils come from and why different for different orders of nuns?

At catholic school, one nun was a servite nun and when teaching geography rather than R.E. on a hot summer's day, she would remove her veil as she claimed she didn't need it as she was not teaching religion to us. But she put it back on if a priest was in the building. A boy asked why - what is he - a king? She said that was the rules. The other nun we had was a Dominican and in her order of nuns, they never removed their veils. Why are different orders different? Also, why do some nuns wear that white garment that goes around their face also and covers all their hair? This seems to ressemble the islamic custom of hiding all their hair. Also, the nuns I knew had had to have their hair chopped off before being allowed to wear the veil. Why is this?