Negative philosophy is Schelling's term for any purely rational idealism — a system that begins from concepts and derives reality from them by logical necessity. Hegel's Logic is the supreme example: it moves from pure Being through the entire categories of thought and arrives, by pure conceptual dialectic, at the Idea as the comprehensive whole of rational reality. But Schelling's objection is decisive: no matter how complete the system of concepts, it tells you only what must be the case given these concepts — it cannot establish that anything actually exists. The concept of God, no matter how fully developed, does not entail that God exists; the concept of the world does not entail the existence of any particular world.
Positive philosophy begins from the other end: not from what must be, but from what is. It begins with the brute, unjustifiable fact of existence — the sheer that of being — and then asks what this fact reveals. God, on this approach, is not the conclusion of an argument but the presupposition of philosophy: the being whose existence is prior to and cannot be derived from any essence or concept. Existence precedes essence — in God absolutely, in finite creatures in a weaker sense. This is the decisive inversion of rationalism.
Kierkegaard, who attended the Berlin lectures and initially hoped Schelling would deliver philosophy from Hegel's abstractions, eventually found the positive philosophy disappointing in its execution — but the distinction between existence and essence, the priority of the actual over the conceptual, went straight into his own work. Heidegger's question of Being as distinct from the being of particular entities, and Sartre's famous reversal of the scholastic formula ("existence precedes essence"), both trace lines back to the positive/negative distinction Schelling articulated in 1841.
The distinction between positive and negative philosophy is the central theme of the Einleitung in die Philosophie der Offenbarung (Introduction to the Philosophy of Revelation), delivered in Berlin in 1841 and published posthumously. Kierkegaard's notes from the lectures survive.