Hey all, photographer here (and web developer/designer - two professions that require a fair amount of knowledge about color). Let me explain what's happening with the image as a whole.

Here is the real dress, or something like it, shot appropriately.

And here is the image in the OP, with some adjustments.

When dealing with light, we need to worry about something called color balance. Basically, a lot of white light that we deal with isn't truly white - that's why there are different types of bulbs (daylight is typically the closest to real, balanced white, but a lot of people see it as blue). The sun covers the spectrum of light that we can see more or less fully. An incandescent bulb burns pretty yellow. The orangish sodium lamps used on street lamps actually only put out that specific wavelength of orange, which is a real bitch as a photographer.

Anyways, if a camera has it's white balance set to auto, as most camera phones always do, the camera needs to guess what type of light it's encountering. This is a measure of blue vs yellow and to a lesser extent green vs pink. The dress being blue completely screwed up this measurement, so the dress isn't as blue as it should have been. This added yellow to the whole image, including the blacks. On top of that, the image is a little over exposed, causing those orangish blacks to look more orange.

Why are some people seeing the dress as white? Their own vision is a part of it, but I would guess a lot of people are on monitors that aren't so great that have brightness cranked. That could make it look something like this extreme example.

Tada.