Short Version: The subject is simply that which experiences and the object is experience itself. The subject is not accessible to experience and is inferred. Objects are everything that can be known.
First, the subject/object distinction and then an elaboration on the Schopenhauer quote.
The subject/object distinction... When I experience things, none of them are "me". Obviously things like cars, chairs, etc... are not me, but even more "personal things" I experience are not me. For instance... My body is an object of experience (representation). My thoughts are also representations. Even my "felt" sense of self is a representation (combination of thought and feelings). I have no access to "myself" except through inference. I infer that with experience there must be one who experiences, and this must be "me". In this sense, the subject (me) is transcendental; it cannot be part of my representation and every time I think I apprehend it, I simply apprehend another object. It must always remain immune to direct experience.
An elaboration on the Schopenhauer quote...
He's elaborating on his claim that reality IS our experience (The World is My Representation [or Idea]). Let's walk through this.
Everything I know, I only know because it is part of my experience (representation). What I don't know simply doesn't exist for me. Yet, since all I can know is what enters my experience, what doesn't exist for me simply doesn't exist.
For instance, take the planet Pluto. It exists for me in the form of representations (experiences reading about it, ideas in my head right now about it, etc...). However, had I never heard of Pluto, then it simply would not exist for me. Saying it exists outside of me is fine and dandy, but then we're talking about the representations of people outside of me. To me, it still wouldn't exist.
Even objectivity is representation (subjectivity) writ large. When I draw the line between what's "real" and "unreal" I do so also through reconciliation of representations. If I see water, I decide if it's really there by reconciling it with other representations. Here is a representation of water, and here is a representation of others saying they see no water. I then weigh the two representations and decide whether the water "really exists". Even that "really" is simply a decision about what future representations I can expect of the water (the experience of it quenching my thirst, etc...).
When I die, for all intents and purposes, the world dies with me. That it continues is an abstraction that has no bearing on me when I die, for I (and my representations) will cease. Insofar as the world exists without me, it too is a representation in my mind of an imagined future in which I'm gone and others still go on.
The world IS my representation.