Summarize:

You can do this one of two ways:  Scroll through your apps pages to locate the Kindle app, which has the black silhouette of a person reading against a blue background. From your home screen, Slide your finger to the right to open your search function. Tap on the search bar, type Kindle in, and tap on the app when it appears in the menu below. Find this icon (three horizontal lines) in the very top, left-hand corner of your screen.  If you are using an iPad, skip this step. If you have a book open—whether on an iPad or iPod—press the settings menu followed by "Library". This will return you to the main screen. If you are using an iPad, tap on the “Settings” icon. Find this icon (a white gear) in the very bottom, right-hand corner of the screen. This will bring you to the “Deregister” page. You will receive the following notification: “Deregister will remove all downloaded content from this Kindle, do you want to continue”. If you register the device with the same account, all of your downloaded content will return to your library. You will be effectively “signed out” of your account. Your device’s connection to the Amazon account it was registered under is severed. Tap “Email or mobile number” and enter the email address or mobile number associated with your Amazon account. Tap “Password” and enter the password for your Amazon account. Press Sign In. Your device will automatically register with the Amazon account you entered.
Launch the Kindle app. Press the Menu icon. Scroll down to the bottom of the menu and press “Settings”. Locate the “Registration” section and click “Update”. Click Deregister this Kindle. Click “OK”. Sign in to the Kindle app to register your device.