Green screening is the process where something (or someone) is filmed against a green background (like this image) and then using a sophisticated software one can switch this green background with anything he likes, an image or another video. So the subject appears as if it was filmed against that new background instead.
Green screening is not a new process. It was done before computers were introduced into the TV industry. But back then, it was A LOT harder to do. Here is a video explaining how it was done!