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:

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:

Although I am not an expert in embryological development, I have tried to find some interesting articles regarding fingerprint development in the womb.

This one outlines the basics about fingerprints.Another website here, has several pages describing fingerprints and the ridge development. To summarize, fingerprints are fully developed by the time a fetus is 6 months old, and These epidermal ridges are caused by the interface of the dermal papillae of the dermis and the interpapillary pegs of the epidermis. The ridge pattern development not only depends on genetic factors but also unique physical condition, such as fetus's exact location in the womb as well as the density of the woman's amniotic fluid. This can explain why identical twins have the same genetic code but can have different fingerprints. Also, I found it interesting that there are a couple of medical conditions, such as adermatoglyphia, in which people actually do not have fingerprints. wikipedia also has some general information about fingerprints.