A lot of people are currently arguing about the colour of this dress, and trying to figure out why they see it differently

Some people see it as black and blue, others as white and gold. A few are making their case using RGB colour pickers to try to prove that it is one colour or another. Counter-intuitively though (and in contrast to what you might have been taught) there is not a simple correspondence between the physical emitted light from an object and its colour. Instead the emitted light features the combined effects of both the properties of the object and of the illumination in the scene (by using the context of the whole scene). Your brain attempts to subtract out the effects of the illumination in order to give you "colour constancy" (meaning that colours are perceived the same in different lighting). This video explains it:

Note that I am not an expert on colour vision, but I did spend a few years teaching students from this video while I was working as a TA. Edwin Land's experiments in this video demonstrate colour constancy in action (and why the colour-picker approach is invalid), and are explained by his retinex theory. Here is the Wikipedia article (which you should take with a heap of salt, but it's better than nothing). I believe what we see with the dress photo is an example of different peoples' brains coming up with different assumptions about what the illuminant is in the photo.

Update 12th March 2015: There has been a lot of semi-private discussion on this topic between vision scientists since the story first came out and many people have written great summaries which go more into why we might see these individual differences. Christoph Witzel does a very good job here. There will be a special issue of Journal of Vision on this topic next year, by which time it may have been all figured out.