Baronetcy is the rank of a baronet. Baronets, such as Sir Walter Elliot, are given status based on their prosperity. Because such a title is based on wealth, it is passed down to the previous Baronet’s heir. Baronets had been given separate titles for Ireland, Scotland, and England, but by this time, only those of England were receiving new titles.
It is not shown whether Sir Walter had acquired his baronetcy in his life or inherited, but it is because of it that he has substance to his character, and losing it is the thing he fears the most. The baronetcy is his title and proof of his wealth and status.