WebDriver Headless Browser is the concept of running web browser based tests but in an invisible browser. When you run a test using Selenium WebDriver a browser window is opened in which you can visually see your http calls being made as well as interactions with your given web page. When running in a headless state your tests still run but the web browser page does not open.
WebDriver Headless Browser Video Tutorial
There are many reasons why you may want to run your tests in a headless state. For instance running your tests in a CI environment such as Jenkins where you wont see a physical instance of a web browser.