In Ch. 7, there is a line about a dispute concerning the right of property of a pig. What is this talking about?
The line just preceeding this one gives the answer: (I will modernize it for your benefit) "At this time, being a simple time, cases of public interest of far less significance than a single woman having a child with a hidden father, often became matters for high courts to decide and may have resulted in the making of new laws." What they then said is that a "dispute about who owns a pig might become a bitter fight in local courts (colony) and then create problems at the level of state government (law).
This is important because the governor is meeting with local religious leaders (it's a theocratic system of law) to discuss Hester Prynn (and Pearl) and to deliberate on taking the child away from it's mother.
He wants to impress upon readers that "law" at that time was quite "new" and that it is much, much different than today.
This was true for crimes of a "moral" sense too.
Adulterers today might face some public ridicule and maybe a divorce, back then they might cut off your nose or ears, whip you, or even put you to death.
You are doing a close reading of the text to notice this--good for you!