In the Hebrew mind the nephesh was the breath which animates the body; it's what distinguishes something which is alive from something which is dead. In Genesis 2 God fashions Adam from the dirt of the earth, breathes into his nostrils and Adam becomes a living creature (לְנֶפֶשׁ חַיָּֽה l'nephesh chaya), a living, breathing, thing.
The Greeks had different ideas, the Platonic concept of the soul and the Aristotelian concept of the soul, for example, were quite different; but generally involves some concept of the animating principle. For Plato this animating principle had its own existence apart from the body, having originated in the pre-mortal world; for Aristotle the soul referred to the innateness of a thing, a knife which cuts is a knife because it cuts, that is it's soul; a tree is a tree because of its innate treeness.
As used in 1 Corinthians 15 to describe the present body (soma psuchekos, "soulish body") it indicates our present mortal life, the way we currently live in our present mortality and corruptibility; in contrast to the future spiritual body (soma pneumatikos) of the resurrection which is immortal and incorruptible.
I would argue that what is aimed at when talking about the soul is the fact that we aren't merely compositions of matter and bio-chemistry, we are more than merely the sum of our parts. We're alive, and more than that, as human beings our life is a kind of life which can relate to God. This is what philosophers and theologians have aimed at in talking about the rational soul; that the rational soul of man means the capacity for reason, consciousness, and conscience--and by which we might confess God.
Is this something independent from the body? Yes and no. David looks forward to reuniting with his dead child in She'ol (2 Samuel 12:23); and King Saul meets the shade of the Prophet Samuel when he wrongly and foolishly seeks out the medium of Endor (1 Samuel 28:8-14). And we have the promise that absent from the body we will be present with the Lord (2 Corinthians 5:8). So in that sense, something about us survives death, at least in some sense; which is why Christianity historically believes that the soul survives death in the interim between death and resurrection, either in the Lord's presence or not. But to think of the soul as some separate thing than us, or that we are merely souls inhabiting a body is completely wrong. Bodily existence is the way things are supposed to be, death is not how it ought to be, and so when death tears us asunder this is to be understood as a deep wrong--which is why God in healing the world brings resurrection.
-CryptoLutheran