Ah - the leftover energy of a soul. When you collect a soul, you're sort of "cutting" is essence out of an energetic shell, and that shell is the Remnant.
It's our main power source among the dead, though it can also be used for various purposes based on its color... for example, a Red-colored Remnant could be used as the heating element in a little, personal heater.

[...]

At least, that's what I know of it. That's more, ah... Violet core territory, as it delves into technology and the like...