Our fingerprints are totally unique, not even identical twins share the same fingerprints! How they form is very interesting and you are spot on, fingerprints develop in the embryo before a baby is born. A person's fingerprints are formed when they are a tiny developing baby in their mother's womb. Pressure on the fingers from the baby touching, and their surroundings create what are called "friction ridges", the faint lines you see on your fingers and toes. These ridges are completely formed by the time a fetus is 6 months old, that's 3 months before the baby is born!
The following website is a fantastic resource on how fingerprints develop and also how they are used to identify criminals:fingerprints
You are absolutely right, the key to fingerprint formation lies in the field of developmental biology. In fact, since our fingerprints reflect the environment we encountered when our life began, some scientists believe that a person's fingerprints may actually be able to tell us about their overall health! You can read the following article for more information:more on fingerprints