SharedPreferences is mainly for application-specific settings that you can access via your Settings menu - like application settings. It's a good idea to keep everything simple here - mostly boolean flags, short strings, or integers. SharedPreferences data persist on device reboot, and are removed along with app uninstallation. Data is saved as a key-value pair.
Internal Storage is mostly used for larger non-persistent data storage. You utilize internal storage if you want to process an image, a short video clip, a large text file, etc. But you don't store the processed data in the internal storage - its function is more like a CPU's RAM. The amount of available internal storage for your application depends on the device, but it's always a good idea to keep anything under 1MB. Data is referenced via its file path.
External Storage does not only refer to the SDCard storage, but for higher-end phones, this can mean internal mountable storage (like in the Galaxy Nexus or S2). This is where you store the large video files, the high-resolution images, and the 20-megabyte text file you want to parse in your application. This is also a place to store data that you want shared across devices if you swap sd cards. Data is also referenced via its file path.
SQLite Databases is where you'd store pretty much anything you want in a regular database - with the advantage of organizing things into tables, rows, and columns. It works best with things that you want displayed in the UI as lists - the best example would be the great invention knows as the CursorAdapter. Data stored here also persist on device reboot, and are removed with app uninstallation. You can also share data across applications with sqlite db if you hook it up to a ContentProvider. Data is accessed using a Cursor, where you can call methods as if you're executing sql statements.
Network Connection is not really a data storage technique, but can be a way of persisting data for a specific user provided the device is connected to the internet, using some sort of authentication. You have to balance out between downloading data every time the app needs it, or having a one-time data sync, which would ultimately lead to another of the storage options mentioned above.
onSavedInstanceState
is also a data persistence technique in Android though it doesn't survive doesn't app or phone restarts. It can be used to restore activity state when activity gets destroyed and restarts later. Few details - How to use onSavedInstanceState example please & When exactly are onSaveInstanceState() and onRestoreInstanceState() called? – Wedding