I am going to try and state this in simple terms, it is more complicated.
Humans have 23 PAIRS of chromosomes. When the egg (or the sperm) , the pairs split into 2, so the egg (or sperm) contains only 23 chromosomes. When they combine in fertilization, then the result is 23 PAIRS again.
How the genes express themselves determine the result that you can see in the offspring. Some genes are dominant, some are recessive. Let's just say, for grins, that the gene for brown eyes is dominant and the gene for blue eyes is recessive. (It is really more complicated than this, I believe there is more than one gene that determines eye color).
If the offspring gets the brown "B" (dominant) gene from both parents, then it will have brown eyes.
If the offspring gets the "b" blue (recessive) gene from both parents, then it will have blue eyes.
What happens if the offspring gets one of each? Since the "B" brown is dominant, the offsprings eyes will be brown, because that gene is dominant over the "b" blue (recessive). Even though the offspring has brown eyes, it will be a carrier of the "b" blue gene and can pass on that gene (or the dominant "B") to it's own children. Depending on what gene that offspring gets from the other parent, the brown-eyed person can end up with a blue-eyed child.
The Punnett Square illustrates the probability of certain offspring occurring.
Basic Principles of Genetics: Probability of Inheritance