Hume's Enquiries are more or less a repackaging of the material from Books I and III of his earlier A Treatise of Human Nature. Ever desirous of literary fame and dismayed by the lack of interest others had shown for his prior tome, Hume went back to the drawing board and attempted to present his philosophical system in a way that would be palatable to the reading public. We should feel fortunate that he did so. For, though the significant changes are in style and emphasis rather than substance, these books are a perfect introduction to Hume's thinking. And while the shorter form did require some not insignificant cutting, most of what you find in the earlier book is presented here in a simpler, more accessible manner. That's not to say that there is nothing new here; there is. In particular, he considers some religious subjects (i.e. miracles and immortality) that he was unwilling to broach in the earlier work.
The connecting thread here is an emphasis on grounding philosophical inquiry in an empirical account of human nature, and particularly of the human mind. The first Enquiry is an account of Hume's take on the implications of the classical empiricism he inherited from Locke and Berkeley. For Hume, as for the other classical empiricists, empiricism was primarily a psychological theory about the origin and content of our concepts. (So empiricism, Hume thought, is a crucial element of any plausible account of the human mind.) The central tenet of this theory is that our concepts are furnished by experience, which includes both sensory experience and introspection (i.e., the experience of our own mental states). And the empiricists also agreed about the way we can justify our beliefs. Some beliefs are true (or false) in virtue of the ideas they contained, and we can know their truth (or falsity) simply by thinking about them; other beliefs are true (or false) in virtue of how the external world is, and we can know their truth (or falsity) only by drawing on our experiences of the world. According to Hume, all substantial conclusions about the world fall into this second category. That is, the truth (or falsity) of all substantial claims about the existence and nature of things in the external world can be discovered only by checking those claims against the evidence of our senses.
Here we seem Hume wielding this philosophy of mind in order to adjudicate disputes in metaphysics and epistemology. Do you want to know whether something can be known? Then think about the concepts in which it is expressed. Could we come to know this by thinking about the meaning of our concepts? Could we come to know it by going and looking or doing certain empirical tests? If the answer to both these questions is no, then knowledge of this subject is an impossibility for us. Do you want to know whether some claim of the metaphysicians is true or whether it even makes sense? Consider the concepts they use to express their views. Is there any way you could reduce the content of this concept to some experience? If not, their claims are literally meaningless.
This interpretation of Hume's project downplays his skepticism and emphasizes his professed intentions to provide a positive account of the operation of the human mind that appealed to nothing beyond the evidence of our senses. According to proponents of this interpretation, Hume is most interested in a description of the operation of the human mind. He's describing what human nature allows us to know and what it doesn't allow us to know. Furthermore, he argues that our nature is such that, where it fails to provide us with the resources to acquire the knowledge we might want, it provides us with a natural habit of forming the right conclusions anyway. Even though our nature limits our knowledge of the world, it ensures that we possess the habits of mind needed to make our way in the world. Hume dubs all these habits of mind "custom."
And I think this naturalistic interpretation of Hume's project provides an entry into the views he defends in the Enquiry concerning the Principles of Morals. Again, it's possible to interpret Hume's project in moral philosophy as a skeptical one. The fact that he thinks morality is based in human sentiments show that he is, in some sense, a subjectivist about morality. He doesn't think there is any plausible account of our moral thinking as based on reason or empirical inquiry alone. Morality, then, is more a matter of feeling than a matter of thinking, observing, and reasoning. But, importantly, Hume doesn't think this is indicative of some problem with morality, and so he doesn't understand himself to be undermining ordinary morality. His aim is to expose the groundless pretensions of reason in order to make room for a wholly naturalistic account morality; it's not to show that morality doesn't have a firm basis. For he does not think that morality would ideally be based on reason and empirical evidence rather than sentiment. Rather, he thinks there is a sort of philosophical overreaching involved in trying to base morality on reason or empirical evidence as opposed to sentiment.
But what is the relevant sentiment? According to Hume, it is a general sort of benevolence, of concern for others. Our possessing such a feeling does not mean that we'll always set aside our own interest in the interest of others; nor does it mean that we are not largely self-interested. It does, however, mean that we're not wholly self-interested, as we are motivated to do (and not do) certain things even when they do not affect our own interests and desires. But what inspires these sentiments, and how exactly do they translate into moral judgments? Morality, Hume argues, is based on sentiments of approbation and disapprobation that are prompted by a recognition of the connection between human actions, dispositions, etc. and what is in the best interest of oneself and of mankind in general. What we take to be virtues, Hume argues, are those dispositions that lead a person to perform actions tending to promote his own happiness and the happiness of others, whereas vices are dispositions that do the opposite.