The File and Directory Entries API simulates a local file system that web apps can navigate around. You can develop apps that can read, write, and create files and directories in a sandboxed, virtual file system.
The original File System API was created to let browsers implement support for accessing a sandboxed virtual file system on the user's storage device. Work to standardize the specification was abandoned back in 2012, but by that point, Google Chrome included its own implementation of the API. Over time, a number of popular sites and Web applications came to use it, often without providing any means of falling back to standard APIs or even checking to be sure the API is available before using it. Mozilla instead opted to implement other APIs which can be used to solve many of the same problems, such as IndexedDB; see the blog post Why no FileSystem API in Firefox? for more insights.