Note 1: (seminary and establishment)
The American Heritage dictionary defines seminary as: A school of higher education, especially a private school for girls.
The dictionary defines establishment as: A public or private institution, such as a hospital or a school. Institution is then defined as: An established organization or foundation, especially one dedicated to education, public service or culture.
The disctinction between seminary and establishment seems to be that an establishment is something grander in its management and scope. Mrs. Goddard's school, therefore, is less than an establishement and less than a seminary.
Note 2: (fancy-work)
The American Heritage Dictionary defines fancy-work as: Decorative needlework, such as embroidery.
Note 3: (natural daughter)
Natural Daughter: This refers to Harriet being an illigitamate child. It is only known to Mrs. Goddard what class her father belongs in, whether a gentleman or in trade.
Note 4: (short, plump, and fair)
Jane Austen rarely gives complete descriptions of her characters, but she was unusually precise in her depiction of Harriet Smith.