Humans, like most familiar plants and animals have paired chromosomes, and are described as diploid. One chromosome of each pair is contributed by each parent, one by the female parent in her ova, and one by the male parent in his sperm, which are joined at fertilization.
The fundamental reason for eye and hair color is all down to dominance: Whether or not the colors are of a dominant or recessive gene. The dominant trait will determine what color the gene will be.
The thing with brown hair and red hair is, that brown is a dominant trait, but red hair is actually a different allele, a variant form of gene. Getting red hair is a recessive trait; you have to get a gene from both parents. A recessive gene is also one that will only express itself when there are no dominant genes present. However, if a person with red hair has a baby with someone who does not have red hair, say a blond, the baby could have blond or brown hair because the red hair trait would have no affect without its counterpart.
As for eyes, it depends on what you are starting with. Brown is dominant over green and blue, but green is dominant over blue. There are actually four different genes on two different chromosomes that code for eye color, so that's why we all have varying shades of colors.
In any case the most important thing is that your baby is born strong and healthy; no matter what color their hair or eyes.