|MadSci Network: General Biology|
As I see you are an undergraduate I'm going to pitch this at quite a high level. If there are any terms or concepts you have trouble with, you can find most of them in any good developmental biology textbook (for example, Scott Gilbert's 'Developmental Biology').
For a good visual guide to eye development, go to The Development Of the Eye Site.
The colour of a person's eyes are dependent upon a structure called the iris. This is a little muscular structure that can contract or relax to control the size of the aperture into the eye (pupil). It is unusual for a muscle in that it derives from ectodermal components of the embryo, rather than being mesodermally derived like most muscles.
The colour of the iris derives from the presence, and number, of pigmented cells called melanocytes. These cells migrate from the neural crest during embryonic neurulation and make their way to their final destination where they differentiate and start producing pigment.
In a newborn infant, not all the neural crest cells have yet found their way to the iris and started producing pigment. Therefore the retinal pigmented epithelium (RPE) portion of the retina (whose melanocytes are derived from the neuroectoderm not the neural crest) can be seen through the iris.
So when you look into the eyes of a newborn, you basically see the dark grey or bluish tinge of the retina itself. It can be up to six months before all the cells become established and produce pigment, and the iris takes on the characteristic colour it will have for the rest of that person's life.
Footnote: As I was searching the web for a suitably helpful visual guide to eye development, I came across some brief notes from a paediatrician about minor ethnic differences in eye colour at birth. I had not heard of this before and cannot speak for its authenticity. However, it would be interesting if there were differences in migration and/or maturation rates for the iris-bound neural crest cells?
As an interesting adjunct to this question, you might want to look up information on albinism. This is a partial, or complete, lack of pigmentation in humans and other animals that is generally caused by failure of the melanocytes to produce melanin (it can also be caused by failure of neural crest migration). If you are interested you might want to start your information gathering at the Facts About Abinism page by King et al. A search of Google for 'albinism' will also generate lots of hits, including many weighty academic pieces.
Hope this helps!
Try the links in the MadSci Library for more information on General Biology.