That you can operate the software without the normal i/o devices (monitor/screen, mouse, keyboard etc).

With browsers, it's commonly used to programmatically take screenshots of web sites.

"headless" - Without a display. Most programs which need an X display, can be run on Xvfb

"completely headless" - Xvfb is also not required.

As previous comments state, you don't need a display, which is a bit of a half truth. Some instances, like running headless Selenium testing with Firefox, would still require a virtual screen to draw to. We do some our browser testing this way, on servers that don't have a screen.