Public Companies are listed on the stock exchange. They are required to release detailed information on a quarterly basis. They are easier to research.
Subsidiaries are companies whose voting stock is more than 50% owned by a controlling (or parent) company. As parent companies are not required to report on each of their subsidiaries these can be challenging to research.
Private Companies are not listed on the stock exchange. They are not required to release information about themselves publically and can be challenging to research.