I don’t think there’s any mainstream religion out there that doesn’t assume the existence of a soul, nor that there is anyone out there who would need the explanation of what is meant by it. Still, I’d say that the soul is nothing but a fancy name for consciousness. And that definitely doesn’t have to be something breathed into us by some higher spiritual existence. Even less does it have to be something unique to humans, however painful that is to certain people to hear. Remember that only a few hundred years ago (and I daresay deep within even today), “civilized people” still refused to consider others humans just because the color of their skin.
It all comes down to learning. Learning is the means by which something adapts to its environment. Something – even “primitive” animals show signs of learning, some animals even use tools, and definitely every single life form on the face of Earth constantly “learns” to adapt to its environment.
As a life form learns more and more, so does its behavior get more and more complex. If someone takes the trouble to observe animals long enough, it can be seen how they make decisions based on their previous experiences – just like humans.
What makes humans different then? Nothing, really. It’s simply the scale at which we are capable to learn that’s extraordinary. Humans developed means to pass on information about experiences and behavior more efficiently than direct experience. Eg you can tell a kid that if this and this happens, then act a certain way. The child can understand that and capable of applying that theoretical knowledge when that condition actually arises.
This abstraction of information and the communication of intelligence is something that is largely unique to humans and it’s what gives us the upper hand against the rest of the world. While animals have to learn from their own experiences, we can learn from the entire accumulated knowledge of our history.
With huge amounts of information comes a great complexity of behavior, something that you can see in human civilization, culture and society. It is entirely feasible that our consciousness is nothing but the most efficient method (selected by evolution) of dealing with information. I don’t think a “soul” has to be fundamentally different from the behavior patterns observed in animals – it’s just an extremely complex behavior pattern (cf chaos theory).