That's a good question.
It's dangerous to store a user's password in a standard local directory on a device, for the obvious reason that if the phone is compromised a hacker may have access to a password that is likely shared between accounts (do you have a different password for every service you use?).
However, storing a username to the device's default storage is not-so-problematic, and that is generally the method of choice. For iOS this would be NSUserDefaults
.
Now, in the case of passwords and tokens (which are certainly necessary and FB would not cut corners on having token-based auth), both being secure contents that ought to be protected, they are generally stored in some sort of encrypted keychain. In the case of iOS, 256-bit encryption by virtue of Keychain Services.
Therefore, when you build an application with auto-login you retrieve the password and token from the keychain on load. However, if the device were to be lost and end up in the wrong hands all of this data would be encrypted and inaccessible.
Of course, let's not pretend this method is fool-proof: http://arstechnica.com/security/2015/06/serious-os-x-and-ios-flaws-let-hackers-steal-keychain-1password-contents/.
EDIT: Although my background is iOS, I am aware that Android uses Keystore as their alternative.
https://developer.android.com/training/articles/keystore.html