In 1800s England, society was built upon a system of primogeniture, a word that directly translates to first-born. In England, it was specifically the first-born son that was meant to be the heir for a man’s wealth.
Persuasion’s core inheritance issue arises from the fact that Sir Walter Elliot has no direct heir; all of his children are female. Because England has a first born male inheritance, or a male preference primogeniture, the next in line would be any nephews of the owner. William Elliot is the eldest nephew, so he is set to inherit the wealth.
This passage’s dilemma is that Sir Walter might have a new wife, and with a new wife comes the risk that he would have a son. A son of any age would take precedence as an heir over a nephew such as William, so he sets out to break off what might become a marriage to secure his wealth.