There is a long-standing popular belief that the differences in eye color are explainable by a single locus difference, where brown eyes are coded by a dominant allele and blue eyes by a recessive allele. This does not explain the gradations represented by green and hazel, for example, and does not conform with examples as the one given in the question.
While the physiological details still have to be worked out fully, a scientific consensus has been reached for many years, that pigmentation in general and eye color in particular are expressed as a polygenic trait, that is, as a result of the action of several genes.
Let us say, applied to the posed question, that two pairs of genes encode eye color: A/a and B/b. A and B enhance pigmentation, a and b reduce it, such that one obtains the following scale:
Blue means either aabb or aaBb;
Green means either Aabb or aaBB;
Hazel means AAbb;
Dark brown means A-B- (that is, AA or Aa and BB or Bb)
Now let us suppose that your parents are:
Aabb x aaBb (father green eyes x mother blue eyes)
By straightforward Mendelian analysis this produces a progeny composed of 1/4 AaBb (dark brown) 1/4 Aabb (green) and 2/4 aaBb or aabb (blue). Incidentally, this 1:1:2 distribution is exactly the one described for this family, but of course a progeny of four could be anything, due to chance.
What does the model tell us?
- That there is a redundancy of genotypes for each phenotype class except hazel, which is commonplace with polygenic characters
- That A and B complement each other, such that all genotypes A-B- are lumped into the same phenotype class (dark brown)
- That A without B has an additive action (compare aabb through AAbb) that is stronger than B without A (aaBb still "looks" blue, aaBB is green while AAbb is hazel).
Of 9 possible genotypes, 4 are for dark brown. Then, why so many human populations have a majority of people with light colored eyes? The answer to this, is the frequency of each allele. In this model, the Hardy-Weinberg distribution (without gametic disequilibrium) predicts a mere 13% dark brown eyes if the allele frequencies for A and B are both 20%, for example (the same example predicts 61% with blue eyes).
Is this fiction? Surely so, it is just a model, an abstraction that is devised to provide a plausible explanation. Most probably it is too simple to work for some or most of other families. But one can argue that it is not too far from reality, and illustrates the polygenic concept pretty well. As said above, the physiological details still have to be worked out: how many genes are really involved, whether they are the same (or analogous) among populations, or how they interact among them to produce the known eye color phenotypes.
Furthermore, at least part of the genes coding for eye color participate in the basal skin pigmentation and hair color as well. But it is believed that this is only a partial overlap, hence green eyed people with black hair and brown-eyed people with blonde hair, etc..
Your parents could have had a gene for their eye colors and another for any other eye color. Meaning there is a probability of you having brown eyes, therefore you ended up with the other eye color.
I am also a dark-brown eyed person born to parents with very light colored eyes- my father has very light brown and my mother light green/hazel. My siblings are a range of light blue to brown. It is because eye color is not dependent on a single set of alleles. Many factors contribute to eye color. I suggest you checkout the link I attached. It really is very interesting!