In high school I learned that blue eyes were a simple recessive gene so two brown-eyed parents could have a blue-eyed child but not vice versa. Recently I discovered that this was wrong. Many genes are involved in eye color and with the subtle variations in eye coloring.
So I ran the fun eye color calculator at GEDmatch.com and discovered that it shows many of the SNPs involved in my eye color with the details of what they each do.
I had not realized that blue eyes are caused by the lack of (brown) melanin, so really are no color. Blue happens the same way the color of the sky does: light is bounced around in a way that appears blue according to the wikipedia article on eye color. Maybe that is why my eyes appear greenish when I wear certain green shirts and turquoise when I wear those color shirts.
If the two blue-eyed parents have their lack of eye color melanin caused by different genes then they could have a brown-eyed child. I happen to know a brown-eyed child of green and blue-eyed parents (and she looks just like her Dad so not what you are thinking). This case got me curious about how that could happen. So I googled around and found this wonderful article that explains it: http://genetics.thetech.org/how-blue-eyed-parents-can-have-brown-eyed-children
Here is my attempt to simplify the ghist of that article:
The two most important genes for eye color are OCA2 and HERC2; if either one is turned off you get blue. So when, for example, a blue eyed man with OCA2 turned off marries a woman with HERC2 turned off, they can both be carrying a brown-eyed gene still and could therefore, in theory, have a brown-eyed child if they each pass on a working copy of OCA2 and HERC2. However this is really unlikely because these two genes are next to each other on chromosome 15 and so tend to get passed on together. In other words, if you have one blue-eyed gene then you usually have both of them. This is an oversimplification and please note that OCA2 has more functions than just eye color.
And of course there are many variations on lighter eye color: green, gray, hazel, and so forth but these colors seem to be just variations that are in addition to the genes making blue eyes.
Here is an article about the origin of blue eyes http://genetics.thetech.org/ask-a-geneticist/origin-blue-eyes
And SNPedia talks about that most important SNP for blue eyes, rs4778241, the one that led perhaps to the simplistic single recessive gene view, since it needs to be a CC at http://www.snpedia.com/index.php/Rs4778241. It is part of the OCA2 gene which is the on/off switch for melanin production. If you are tested at 23andme, you can look at what you have on rs4778241 by clicking here. The important SNP in HERC2 is Rs12913832 – with a GG there you get blue eyes 99% of the time: 23andme tests this so click here for your result.