Some say it has to do with emotions or the structure of your brain. That is not the case. If you look at the photo, you will notice a bright light source in the top corner.
When I first looked at the picture, I saw it as I would see something in front of me. I decided the bright light would heavily illuminate anything in front of it, and so I automatically corrected the colors slightly darker. All of this happened without my being aware of it, because it would happen if I was looking at a well-lit dress irl, and so I saw black and blue.
When my father first looked at the picture, he saw it from the perspective that it was not real, but a picture. He assumed the light would make the camera lens close up and not absorb the colors as well, so he automatically corrected the colors slightly lighter, again without being aware of what he was doing. He therefore saw it as white and gold.
After some research online, we discovered an article which had applied digital editing to the photo. When corrected for excess brightness, it turned black and blue. When corrected for being washed out, it turned white and gold.
Finally, we looked for a photographer's perspective. The general consensus seems to be that the light source in the photo would, in fact, have caused the camera to close up and take in less of the dress's color.
I saw black and blue because I was looking as though it was in front of me.
The dress, in reality, is white and gold.