In Britain such a person is called a midwife. These are nurses who have done additional training in 'midwifery'. They care for mothers-to-be before, during and after the birth of the child and give advice on everything from morning sickness to breast feeding. They are also qualified to supervise the mother's labour and the birthing process.
In some countries this work is done by doctors, but in Britain midwives mostly take exclusive care of all uncomplicated births. Obstetricians are often only called upon if problems arise. The practice dates from times when birthing was supervised by wise women. Today it is a highly skilled profession with rigorous training.
I believe that Commonwealth countries such as Canada, Australia and New Zealand have midwives. I am not sure about the United States. Other European countries have similar professionals. In France a midwife is a sage-femme.