Some years ago a correspondent on the UK midwifery e-group asked what the etymology of this word was. I replied that in Tudor times the word 'quick' meant 'living' or 'alive'. If you bite your nails to the quick you are biting them down to the point where they meet the living flesh and the original words of the Apostles Creed in the Book of Common Prayer states that God shall come to judge 'the quick and the dead'.
So quickening is when the mother feels the baby move for the first time and this movement is a demonstration that the baby is alive. The mother's perception of the pattern and quality of fetal movement is still one of the best indications of fetal well-being, which I find pleasing because it emphasises the importance of the mother's subjectivity and does not require complex electronic technology. In the days before ultrasound and reliable pregnancy tests it was not a positive sign of pregnancy -- if I remember rightly feeling fetal parts and hearing the fetal heart were the positive signs but it was a suggestive sign.
Quickening, like all phenomena associated with childbirth, is not purely physiological. Barbara Katz Rothman in her book The Tentative Pregnancy described a situation in the United States where women who were anxious about the well-being of their fetus could only test this with an amniocentesis. Amniocentesis can only be carried out after a certain gestation: the uterus needs to be big enough for there to be sufficient amniotic fluid to be drawn off. She found that these women, even if they were multiparous, only felt the baby move after the amniocentesis had been carried out although it would have been expected that they should have felt the baby move sooner, since they were multiparous.
I felt my first baby move at 15 weeks. It was a distinct tapping feeling, utterly unmistakable. With my second child I felt vague flusterings from 13 weeks with again the same distinct tapping sensation at 15 weeks.