By all outward appearances, the academy is a private high school for rich kids. As per usual in The Secret World, things aren't always what they seem on the surface. Innsmouth Academy serves as something of a prep school for children of the Illuminati, "and a degree from Innsmouth Academy opens many doors for a budding magus and occultist."
The school has a lengthy history, and originally opened its doors in 1798 before being rebuilt three times as a result of three different disasters (including the opening of a dimensional portal in the elementalism lab). Head to the official Secret World website to read more about the school and its faculty.