yeah, I think you get the gist of it. For Socrates the soul is what made a person alive. A body without a soul would just be a body and would be dead. The soul gives it birth. It isn't tangible by any means.
However, Socrates would argue that the soul is indeed immortal. When we die, it is only the death of the body-- the soul goes on to the underworld and comes up again to be re-born and take the form of another body. And since it does this, Socrates said that there is no such thing is "learning." Like learning is really recollecting knowledge that you previously had (since your soul had lived before it has already acquired knowledge). I don't really agree with this; but it is how the ancient Greeks saw it.