What does "headless" mean?
"Headless" in this context simply means without a graphical display. (i.e.: Console based.)
Many servers are "headless" and are administered over SSH for example.
Headless means that the application is running without a graphical user interface (GUI) and sometimes without user interface at all.
There are similar terms for this, which are used in slightly different context and usage. Here are some examples.
Headless / Ghost / Phantom
This term is rather used for heavy weight clients. The idea is to run a client in a non-graphical mode, with a command line for example. The client will then run until its task is finished or will interact with the user through a prompt.
Eclipse for instance can be run in headless mode. This mode comes in handy when it comes to running jobs in background, or in a build factory.
For example, you can run Eclipse in graphic mode to install plugins. This is OK if you just do it for yourself. However, if you're packaging Eclipse to be used by the devs of a large company and want to keep up with all the updates, you probably want to find a more reproducible, automatic easier way.
That's when the headless mode comes in: you may run Eclipse in command line with parameters that indicate which plugins to install.
The nice thing about this method is that it can be integrated in a build factory!
Faceless
This term is rather used for larger scale application. It's been coined in by UX designers. A faceless app interacts with users in a manner that is traditionally dedicated to human users, like mails, SMS, phone... but NOT a GUI.
For example, some companies use SMS as an entry point to dialog with users: the user sends a SMS containing a request to a certain number. This triggers automated services to run and reply to the user.
It's a nice user experience, because one can do some errands from one's telephone. You don't necessarily need to have an internet connection, and the interaction with the app is asynchronous.
On the back-end side, the service can decide that it does not understand the user's request and get out of the automated mode. The user enters then in an interactive mode with a human operator without changing his communication tool.