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: