LocalStorage and IndexedDB are storage facilities in your browser on your device. They are established web standards and supported by Firefox, Chromium, Opera, Safari and more.
An example is the local search index in Runbox 7 which is stored in IndexedDB. This makes it possible for you to search email without accessing the Runbox server. No way for any man in the middle or runbox servers to know what you are searching. Will also make it possible in the future to search encrypted emails that you can only open on your device ( a server-based search wouldn’t be able to do this ).