@FUZZYI think he means if you list out their ages in order, there will be no repetition and no gaps. So ages 1-19, 2-20, 3-21, etc.
Sum of ages, I would assume applies to eldest sons, not all males.
In araver's solution, S is the sum of ages, X is the oldest son in the village, and Y appears to be the youngest.
I'm not sure where the y-1 came from. If you don't get that either, here's another way of looking at it using the same 3 original equations. Probably just a slightly longer form of his solution.
@araver Maybe a typo. 10*(Y-1)+10*19 = S = 19*(Y-1) + 19*10
The "10*19"s cancel along with the "(y-1)"s leaving 10=19?