The app category is the category in which an app is published in the app stores, that reflects an app's purpose. Those categories are fixed by the app stores publishers (like Google or Apple) and are meant to regroup similar types of apps to make it easier for users to navigate the app store.
The category ranking is a classification of all apps in category. The higher the app ranks, the more visible and powerful it is on the app store. Some apps don't rank at all, meaning they have very weak visibility. A high visibility is usually tied to a high amount of downloads.