Android: Tap Account then Exit to simply exit the application without logging out. This will enable to you to still receive push notifications about messages and incoming calls but the application will no longer run in the background. Tap Account then Logout to sign out of your account. This will allow you to sign in with a different account and you will no longer receive any push notifications.
iOS: Tap Account then Logout, to no longer receive push notifications about messages or incoming calls. To exit the app, while still receiving push notifications, double-click the device's Home button to see the list of running apps, touch and hold the app icon until the running app icons jiggle, touch the close (-) button on the Fongo icon, then touch the main part of the Home screen or press the Home button to exit jiggle/close mode.