Retrieving passwords that have been used on a phone depends entirely on the model of the phone. There are some universal ways to retrieve data. The cache or temporary files of the phone's browser may keep the user logged in to the site he was on. Some smart phones may have applications that log key strokes, which can then be retrieved in a log file. The iPhone has an app that supports this feature.
Enable history and storing of passwords and cache in the phone's browser settings. This will enable the phone to store more information regarding the password. Not all phones have this option. Some phones will automatically save passwords, if the setting to do so is enabled.
Visit the website the person logged in to. If he did not log out, his account still will be open on the phone. Check his account and password settings to retrieve password information.
Download a keylogger application to the phone. ITrust is an event logger app for the iPhone, it will log and record the past five sessions on the iPhone. Simply enable the keylogger application and allow the person to input his password. The application will record all keystrokes to a log file. Access the log file to see the recorded strokes.