You can use the Firefox developer tools on your desktop to debug code running remotely: in a different process on the same device or on a completely different device. To do this you use Firefox to attach the Toolbox to the remote process, and the Toolbox is then launched in its own window. At the moment the following tools support remote debugging:
Firefox for Android
Remotely debugging Firefox for Android describes how to connect to Firefox on an Android device over USB.
Firefox for Metro
Remotely debugging Firefox for Metro describes how to use desktop Firefox to debug code running in Windows 8 (Metro-style) Firefox.
WebIDE includes instructions for connecting the Firefox developer tools to the Firefox OS Simulator or to a Firefox OS device.
Remotely debugging Thunderbird explains how a combination of Firefox and Thunderbird can be used to debug code running in Thunderbird.