What does it mean to call Hume a "naturalist"?
Hume suggests that most of our beliefs and judgments are grounded in nature, not in reason. If we try to justify our beliefs by means of reason, we will invariably come up short. Our beliefs and judgments are not formed by some rational process, but by the force of habit. We infer that all heavy objects fall to the ground because we have observed countless instances to support this belief and none to contradict it. According to Hume, habit and custom are the only means according to which we can reach any general conclusions about the world around us. Nature imbues us with the instinctual ability to form these conclusions.
What is the distinction between impressions and ideas? To what use does Hume put this distinction?
Impressions are direct and vivid mental sensations. They include, but are not restricted to, immediate sense impressions of sight, sound, touch, etc. Emotion and volition are also impressions. Ideas are secondary in that they are about impressions. They are memories, thoughts, beliefs, etc., concerning our impressions. While most of our thinking takes place on the level of ideas, Hume suggests that all ideas must be traced back to the simple impressions that inform them. Any idea that cannot be grounded in a simple impression is meaningless. Thus, Hume dismisses the bulk of metaphysics as consisting in meaningless ideas.
What is the distinction between relations of ideas and matters of fact? To what use does Hume put this distinction?
Relations of ideas are necessary, a priori truths that cannot be denied without contradiction. Mathematical truths are the best example of this class. No amount of empirical evidence can refute relations of ideas since they do not deal with actual objects, but only with the structure of ideas. Matters of fact are contingent, a posteriori truths that we learn from experience. These can be denied without contradiction: if I say, "it's raining" when the sun is out, I am mistaken, but there is nothing illogical about my assertion. Hume asserts that relations of ideas cannot say anything substantial about the world, so that all facts about the world must be grounded in experience. This serves, first of all, to severely curtail rationalist metaphysics. It also imperils the inductive reasoning we use to connect matters of fact. Inductive reasoning cannot be grounded in experience, nor does it consist solely of relations between ideas.