Who sees not […] that whatever is produced or improved by a man's art or industry ought, for ever, to be secured to him, in order to give encouragement to such useful habits and accomplishments? That the property ought also to descend to children and relations, for the same useful purpose? That it may be alienated by consent, in order to beget that commerce and intercourse, which is so beneficial to human society? And that all contracts and promises ought carefully to be fulfilled, in order to secure mutual trust and confidence, by which the general interest of mankind is so much promoted? (III.II.7)
Would we have the same motivation to work hard if we weren't allowed to own stuff? The idea of property encourages folk to spend their time carrying out useful tasks. Likewise, if a house is passed down in a family, the new owners inherit this motivation. This doesn't mean there's no flexibility: property isn't something we're tied to, and we can exchange it and enter into contracts or promises that are beneficial. Once we've agreed to any promise of this kind, we trust that both sides will keep to the bargain. Why? Altogether now: because it's useful and beneficial.
What other reason, indeed, could writers ever give, why this must be mine and that yours; since uninstructed nature surely never made any such distinction? The objects which receive those appellations are, of themselves […] totally disjoined and separated from us; and nothing but the general interests of society can form the connexion. (III.II.9)
While some virtues/vices are natural, there are other categories that have been created because they serve a purpose. Think about it: the idea of ownership didn't just spring up from nowhere, and it's not something that's always been around. The idea that this is mine and this is yours makes up a social contract that we all recognize. There's no natural link between us and the stuff we own. It's only through shared ideas and language that the idea of property has come about.
The imagination is influenced by associations of ideas; which, though they arise at first from the judgement, are not easily altered by every particular exception […] To which we may add, in the present case of chastity, that the example of the old would be pernicious to the young; and that women, continually foreseeing that a certain time would bring them the liberty of indulgence, would naturally advance that period, and think more lightly of this whole duty. (IV.7)
Though folks may be more laid-back today, chastity has traditionally been classed as a virtue—especially for women. Don't worry—our inner feminists are bristling with disdain, too. But it's true: back in the "good" old days, failure in this area was seen as harmful to both the family unit and society at large. So much for sexual liberation. We could see this virtue as only applying to women of childbearing age, but, as Hume explains, the argument is that older women acting in an immoral way would set a bad example. We're not saying this is true, but it makes for a pretty interesting debate.
But in ancient times, when no man could make a figure without the talent of speaking, and when the audience were too delicate to bear such crude, undigested harangues as our extemporary orators offer to public assemblies; the faculty of memory was then of the utmost consequence, and was accordingly much more valued than at present. (V.I.19)
This is a point that Hume makes again and again: what's seen as useful and agreeable can depend on the society and time period we're talking about. Public speaking was a must for anyone seeking a rep in ancient times. Only the best would do, and audiences were seriously tough. That's why memory was so valued: it was vital skill for any public speaker.
Sympathy, we shall allow, is much fainter than our concern for ourselves, and sympathy with persons remote from us much fainter than that with persons near and contiguous; but for this very reason it is necessary for us, in our calm judgements and discourse concerning the characters of men, to neglect all these differences, and render our sentiments more public and social. (V.II.27)
This quote shows that Hume's a realist—he doesn't imagine we're living in some magical place where everyone's super-caring and nice. It's only natural that our strongest feelings should center on ourselves and those closest to us. What Hume's saying, though, is that it's precisely for this reason that we sometimes need to put aside our bias and think, "hmm, what's best for society?"
In most countries of Europe, family, that is, hereditary riches, marked with titles and symbols from the sovereign, is the chief source of distinction. In England, more regard is paid to present opulence [...] Where birth is respected, unactive, spiritless minds remain in haughty indolence, and dream of nothing but pedigrees and genealogies: the generous and ambitious seek honour and authority, and reputation and favour. Where riches are the chief idol, corruption, venality, rapine prevail: arts, manufactures, commerce, agriculture flourish. (VI.II.13)
Here's another example of culture clash. In Hume's days, status in most of Europe was all about family name and the riches that folks inherited; in England, though, it was more about a person's current wealth. We're not saying one is wrong and the other's right. When it's all about the family line, this can lead to people becoming lazy and entitled (boo!), but it can also lead to ambitious types seeking a reputation (yay!). When it's all about current riches, meanwhile, people can become corrupt and dishonest, but major industries can make a tidy profit. So, what we're saying is there are pros and cons on both sides.
When a man denominates another his enemy, his rival, his antagonist, his adversary, he is understood to speak the language of self-love, and to express sentiments, peculiar to himself, and arising from his particular circumstances and situation. But when he bestows on any man the epithets of vicious or odious or depraved, he then speaks another language […] which he expects all his audience are to concur with him. (IX.I.6)
This quote sums up the difference between the personal and the social. When someone sees another person as an enemy or rival, they've got a private score to settle. However, once they class someone as vicious or immoral, they're not just saying that they have a beef with them—they're making a judgment about this person's character and behavior, and they expect us to nod and say "you're so right!"
Whoever has a high regard and esteem for me flatters my vanity; whoever expresses contempt mortifies and displeases me; but as my name is known but to a small part of mankind, there are few who come within the sphere of this passion, or excite, on its account, either my affection or disgust. But if you represent a tyrannical, insolent, or barbarous behaviour, in any country or in any age of the world, I soon carry my eye to the pernicious tendency of such a conduct. (IX.I.7)
It's human nature to feel good about ourselves when others flatter us or to feel angry/upset when someone criticizes or looks down on us. The thing is, no matter how many friends we may have on Facebook, this is a tiny fraction of the human race. What about all the other stuff that goes on in the world? When we hear about brutality or oppression, even if it took place in a past age or a faraway country, we see it as bad. Sometimes, we have to go beyond our own private bubble to make a moral judgment.
By our continual and earnest pursuit of a character, a name, a reputation in the world, we bring our own deportment and conduct frequently in review, and consider how they appear in the eyes of those who approach and regard us. (IX.I.10)
Here, Hume points out that we don't just spend our days wandering around in our own private world, oblivious to how folk see us. Instead, we look at ourselves through other people's eyes and think about how we come across. This isn't always a good thing; especially if we start obsessing over what others think. On the flip side, though, it's by reviewing our behavior that we get an idea of how others see us and whether there's anything we need to work on.
Thus, two men pull the oars of a boat by common convention for common interest, without any promise or contract; thus gold and silver are made the measures of exchange; thus speech and words and language are fixed by human convention and agreement. Whatever is advantageous to two or more persons, if all perform their part; but what loses all advantage if only one perform, can arise from no other principle. (AIII.8)
Through the simple example of two guys rowing a boat, we get an idea of how useful cooperation can be. Throughout Enquiry, Hume emphasizes that it's natural for people to organize themselves into groups and societies: it's all about people working together to achieve something, and this has led to common languages and currencies. Pretty straightforward, no?