Well... you'll be unsurprised to hear that it is, in fact, more complicated than that.
We'll start with the bit that isn't a lie. Brown eyes are dominant. Even having one copy of "the brown eyed gene" will mean you probably have brown eyes. There are, however, several "brown eyed genes" governing the production and deposition of melanin in the iris, all of which need to work to give you those brown eyes.
So, from what we've learned so far, we can deduce that blue, the "recessive" colour, is the absence of brown, and the colour of an unpigmented iris? Sadly not. A completely unpigmented iris is a pinkish-grey colour deriving from the visibility of its blood supply, and can be found in albinos, ie people with no melanin *anywhere*. Blue eyes come from a genetic mutation that massively downregulates melanin production in the stroma of the iris, ie the layers closest to the surface. The lower layers still contain melanin, which can be demonstrated by dissection and microscopy.
So how does that make eyes blue? The answer is physics, more specifically, Rayleigh scattering, which I am not even going to pretend to understand myself, save that light with longer wavelengths (ie red and yellow) is preferentially absorbed by the melanin in the lower layers of the iris, leaving the shorter blue and purple wavelengths to be reflected and then scattered in the stroma. This is similar to the effect that causes the sky to be blue on a sunny day, and also means that yes, blue eyes *can* change colour according to lighting conditions.
The next colour to look at is grey. It is thought that grey irises reflect and scatter light in a similar way to blue, but in a much less frequency-dependent way (Mie scattering). The best theories we have so far for why this happens is that grey eyes have more collagen deposits than blue eyes, or that they simply have slightly more melanin than blue eyes.
Next, we'll look at the little-known (in humans) eye colour amber. Amber eyes are low on melanin, like blue eyes, but have deposits of pigments called lipochromes in the iris, which is, well, an amber colour. The genetics of amber eyes are poorly understood, probably because it is rare to carry genes that produce amber eyes without also carrying at least one brown-eye gene, which would mask it.
Green eyes are usually a combination of a strong blue colour with either some amber pigment or a little melanin in the stroma of the iris, effectively putting a yellow filter over the blue structural colour to produce green.
Hazel is a catch-all term for greenish-browns, often with concentric rings of different colours caused by different amounts of melanin deposition different distances from the pupil.
And how about violet eyes, the mainstay of terrible Mary Sue fanfic? In the real world, they are a result of incomplete albinism- melanin producing genes that just about work, a little bit, but can't really produce enough pigment to convey proper protection against the sun. Tiny amounts of structural blue colour as described above combine with the red colour of the iris' blood supply to make eyes that appear violet. So if you ever see a Mary Sue with violet eyes but hair that is any colour other than white-blonde, you can now get as annoyed as I do about it :D
It's also worth taking away from this article that blue and green eyed people *still have melanin in their irises*. So while it's rare for two light-eyed people to have a brown-eyed child, it's far from impossible, and should not be taken as more than circumstantial evidence to doubt paternity.
To end on a final, awesome note: I didn't actually know about the amber eye gene(s) until I researched this article. My dad has bright green eyes and my mum dark brown, so I'd always been a bit confused about how I got these strange golden brown eyes that weren't explained in any of the textbooks. But now it seems abundantly clear that my mum is heterozygous for the dark brown gene, and my dad's green eyes are presumably the result of one amber gene over blue. Allowing me to inherit my dad's visible amber gene and (presumably) my mum's invisible one.
http://en.wikipedia.org/wiki/File:HumanFemalewithAmberIris.jpg - eyes like mine!
P.S. Any questions from the floor? I will try to answer them between now and new year.