A collier working by the piece is supposed, at Newcastle, to earn commonly about double, and in many parts of Scotland about three times the wages of common labour. (1.10.18)
In many cases, people's social class will depend on how much skill or education they have. For example, a collier (a dude who makes charcoal) with a special set of skills is more likely to make a good living than someone who can be easily swapped out for the next worker who comes along.
Fourthly, The wages of labour vary according to the small or great trust which must be reposed in the workmen. (1.10.19)
The amount that a person gets paid for a job will always be connected to how much responsibility that person has. If you're the one who controls the main buttons to a nuclear reactor, chances are you'll be paid more money than someone who guards the peanut butter in the break room.
This inequality is upon the whole, perhaps, rather advantageous than hurtful to the publick. It may somewhat degrade the profession of a publick teacher; but the cheapness of literary education is surely an advantage which greatly over-balances this trifling inconveniency. (1.10.95)
In Smith's mind, public schoolteachers probably deserve to make more money based on their skills and hard work. But he thinks that it's worthy to sacrifice their comfort for the sake of making education as affordable as possible to many young students. In this case, he's willing to sacrifice an entire class of workers to the public good.
But, among the ancient Romans, the lands of the rich were all cultivated by slaves, who wrought under an overseer, who was likewise a slave. (4.7.3)
Adam Smith is no fan of slavery, and he's glad he lives in an England where the practice is all but gone. In the old days of Rome, people used to have entire classes of slaves, where some slaves were in charge of other slaves.
The greater part of the citizens had no land, and without it the manners and customs of those times rendered it difficult for a freeman to maintain his independency. (4.7.3)
In the old days of England, very few people owned farmland, which meant that a few landowners pretty much controlled the society and farmers worked for them like slaves. In modern times, Smith looks forward to a world where there is more competition and you have more people owning land and trying to make it productive.
[So] the labour of farmers and country labourers is certainly more productive than that of merchants, artificers and manufacturers. (4.9.18)
Even though business people tend to be more respected, Adam Smith is convinced that there is no one in society more productive than a farmer. That's because these folks produce the food that people need to survive. And better yet, they grow the stuff out of dirt, which seems like they're producing something out of nothing.
Among brothers and among sisters, the eldest always take place; and in the succession of the paternal estate every thing which cannot be divided, but must go entire to one person, such as a title of honour, is in most cases given to the eldest. (5.1.38)
In Smith's time, it was common for all of a family's property to get handed down to the eldest son. That was mostly because people were prejudiced in favor of men over women and because older was always considered better than younger.
All families are equally ancient; and the ancestors of the prince, though they may be better known, cannot well be more numerous than those of the beggar. (5.1.40)
Adam Smith thinks it's silly that some families in England are respected just based on the idea that they have a long bloodline. The blunt fact is that if you go back far enough, we all come from the same families. So the idea that some families are better than others is dumb.
They [birth and fortune] are the two great sources of personal distinction, and are therefore the principal causes which naturally establish authority and subordination among men. (5.1.43)
There are no two things that put a person in the upper classes more than their money and the family they were born into. Nowadays, we care more about money than family. But back in Smith's day, you could be totally poor and still be respected if you were from a good family.
They feel themselves, each individually, more respectable, and more likely to obtain the respect of their lawful superiors, and they are therefore more disposed to respect those superiors. (5.1.93)
Adam Smith is in favor of educating people from all classes because he thinks educated people will try to act like the upper classes. In other words, they'll stop making a fuss about workers' rights.