Bertrand Russell famously summarized Hume's contribution to philosophy, saying that he "developed to its logical conclusion the empiricist philosophy of Locke and Berkeley, and by making it self-consistent made it incredible." Hume is remarkable in that he does not shy away from conclusions that might seem unlikely or unreasonable. Ultimately, he concludes that we have no good reason to believe almost everything we believe about the world, but that this is not such a bad thing. Nature helps us to get by where reason lets us down.
Hume is unquestionably an empiricist philosopher, and he strives to bring the rigor of scientific methodology to bear on philosophical reasoning. His distinction between relations of ideas and matters of fact is absolutely crucial in this respect. Anything we can say about the world is a matter of fact, and thus can be justified only through experience and can be denied without contradiction. Relations of ideas can teach us about mathematical truths, but cannot, as some rationalist philosophers would have, teach us about the existence of our selves, an external world, or God.
If we are left with only matters of fact to get us by in the world, however, we find ourselves greatly limited. How can past experience teach me anything about the future? Even to infer without circularity that future experience will resemble past experience requires some principle that cannot be grounded in past experience. Without that principle, our ability to reason according to cause and effect, and thus the greater part of our ability to reason with matters of fact, is sharply curtailed.
We should be careful to note the tone Hume's skepticism takes here, however. Rather than conclude that we cannot know anything about future events or the external world, he concludes that we are not rationally justified in believing the things we do. Hume does not deny that we make certain inferences based on causal reasoning, and indeed insists that we would be unable to live if we didn't do so. His point is simply that we are mistaken if we think that these inferences are in any way justified by reason. That is, there are no grounds for certainty or proof of these inferences.
Hume is a naturalist because he suggests that nature, and not reason, leads us to believe the things we do. Habit has taught us that we are safe in making certain inferences and believing certain things, and so we don't normally worry about them too much. We cannot prove that there is a world external to our senses, but it seems to be a relatively safe assumption by which to live. Rather than try to justify our beliefs or identify the truth, Hume seeks simply to explain why we believe what we believe.
The Enquiry is decidedly a book about epistemology and not about metaphysics. That is, Hume is concerned about what and how we know, and not at all about what is actually the case. For instance, he does not deal with the question of whether there actually are necessary connections between events, he simply asserts that we cannot perceive them. Or perhaps more accurately, Hume argues that, because we cannot perceive necessary connections between events, the question of whether or not they actually exist is irrelevant and meaningless.
Hume is an ardent opponent of rationalist metaphysics, which seeks to answer questions such as whether or not God exists, what the nature or matter and soul is, or whether the soul is immortal. The mind, according to Hume, is not a truth-tracking device, and we misuse it if we think it can bring us to metaphysical conclusions. A Humean science of the mind can describe how the mind works and why it reaches the conclusions it does, but it cannot take us beyond the confines of our own, natural, reason.