Google allows you to use the Google Play Music service on up to 10 different devices with your account. If you've reached your device limit and want to add a new device, you'll need to deauthorize an old one before being allowed to add another device.
There are two ways to deauthorize devices from your account: using a Web browser or through the Google Play Music Android app. The device management feature was just added to the Android app this week, so we'll start there.
Deauthorize from Android
From your Google Play Music Android app, go to Settings, then select My devices. If you don't see My devices, make sure you're running the latest version of the app. Tap on the "X" next to the device you want to deauthorize, then tap the Deauthorize button to confirm.
Deauthorize from the Web
To deauthorize a device from your Google Plays Music account using a browser, go to Google Play Music Settings. In the Manage My Devices section, click the Deauthorize link next to the device you want to remove, then click on the Deauthorize button to confirm.
That's it. Keep in mind that the iOS version of Google Play Music has not been updated with device management, so iOS users will need to use the Web browser method to deauthorize devices for now.