The color of a person's eyes depends on the amount of a pigment called melanin present in the iris of the eye (melanin is also responsible for the coloring of our skin). At each end of the spectrum you have blue-eyed people, who have relatively small amounts of melanin, while brown-eyed people at the other end have lots of melanin present. People with other eye colors generally fall somewhere in the middle of these two extremes. What determines how much melanin is present in the iris is determined by hereditary genetics.
Simply put, a baby will inherit genes from both of his/her parents and these genes combined will determine the eventual eye color of the baby. We say 'eventual' because all babies (with a few exceptions) start out life with either blue/gray or brown/black eyes (generally speaking, lighter skinned babies with very little melanin start out life with blue eyes while darker skinned babies with larger amounts of melanin will have brown/black eyes). The eventual color of your baby's eyes therefore may change as his or her body begins to produce melanin - and this can take anything up to three years (although for most babies this change will occur from around 6 months of age to one year). Thus, babies who have blue eyes in the early months of their lives may end up having brown eyes as they get older (in fact, eye color in some people can change even in adulthood).
A common misconception is that brown-eyed parents will only produce a brown-eyed baby. This is not correct as it is entirely possible for brown-eyed parents to have offspring with blue eyes (or any other color for that matter). Of course, certain combinations of eye color in the parents will increase or decrease the chances of a particular eye color being produced in the baby, but as with most things to do with Mother Nature nothing is ever 100% certain. Likewise, just because both parents have blue eyes, or green eyes, doesn't mean that the baby will inherit the same eye color. This is because humans carry two copies of every gene, one from the mother and one from the father. These two versions of each gene are called 'alleles' and some alleles in each pairing are more dominant than others. In the case of the genes that control eye color, a brown allele for example is dominant over a blue allele, but a child may inherit the recessive allele from either or both parents.
To illustrate this, consider the large letter 'B' represents the allele for Brown eyes, and a small letter 'b' represents the allele for blue eyes (every individual has a combination of two alleles for eye color, one inherited from each parent). Because we know that brown (B) is dominant over blue (b), a brown-eyed individual can be either 'BB' or 'Bb', while a blue-eyed individual can only be 'bb'.
So to show how a blue-eyed baby can be born to a father with brown eyes and a mother with blue eyes consider the following:
The father has brown eyes, so he has one of two possible combinations, either 'Bb', or 'BB'. Remember that brown (B) is dominant over blue (b), so the blue allele (in the first scenario) for the father is called recessive. The mother has blue eyes, so the only possible combination she may have is ‘bb’. Thus, for the baby to receive blue eyes, the father must have had the 'Bb' pairing and subsequently passed the recessive 'b' gene on to the child. The mother, having 'bb', can only pass a 'b' on to the child. The resulting combination for the baby is 'bb' resulting in a blue-eyed child. If the father had had the 'BB' combination, a blue-eyed child would not have been possible.
In the case of two brown-eyed parents having a blue-eyed baby, both parents would have to be a 'Bb' combination (a dominant brown allele and a recessive blue allele), and both would need to pass the recessive 'b' alleles on to their offspring. If either parent passed the dominant 'B' on to the child, the result would be brown eyes.
Of course, genetics is a lot more complicated than the simplified examples above, and current thinking is that there are a number of genes (not just two or even three) that influence the final eye color of an individual. But whatever the outcome, your baby's eyes will be the most beautiful in the world.