That is a very good question, and one I don't think I've actually seen on /r/AskHistorians yet.

Unfortunately the answer is not something complicated or anything long, or absolutely fascinating, it's quite simple - it was a chemical process that initially began to see use because it slowed down the aging process and made them resistant to environmental factors. It helped reduce fading in photographs, and made them easier to archive, and display, without having to worry about them becoming completely sun-bleached. Later on, the process of 'toning' an image (Altering the silver-based already-exposed emulsion) was sometimes used to darken or lighten images, add a touch of contrast, or to simply give something to an image other than a stark white portrait; a dash of color. This toning process was what gave images their Sepia tint in the case of silver-based emulsions, which is what most photographers preferred.

When an image is Green or any other color than Sepia, it's most likely because a different toning process was used, although it could also mean it was photographed using a different method - it's hard to tell really - but in most cases, it was simply just a means of preservation, I'm afraid. Later on, in the modern, digital age, it's used as a kind of aesthetic. Something you would add if you were trying to instill a feeling of nostalgia in the viewer, whether the image was taken a week ago, or a year ago. It also adds a certain kind of warmth to an image that's hard to find elsewhere. We tend to remember the past fondly, so when we see the old, oft-recognized Sepia tint, we tend to look at it with a little bit of nostalgia in us. I do anyway.

If the image was printed on a metal plate, oxidization over time can change the color, too.