What is the medical term meaning female egg?

The "Ovum". The Word Gamete means the same but is seldom used.
Before the egg cell comes in contact with the sperm cell, it is simply called an egg. Once it has joined with the sperm cell, it is called a zygote. Once the zygote develops certain things, it becomes an embryo, which will soon become your child!

The unfertilized egg is also called an ovum.