A multi-stage ko is a special ko fight, where taking the ko produces another ko mouth. This new ko can then be fought, and this may continue for several stages. A multi-stage ko is also known as an iterated ko or dan ko in Japanese. In the past, a two stage ko was sometimes called a "two step ko" in English.
In a multi-stage ko, the fighting shifts from one ko mouth to another, so that when the original underdog takes the next ko, the opponent is taken one ko fight away from winning the whole ordeal. In a two stage ko that means that the opponent becomes the new underdog.
In a two-stage ko taking the ko produces another ko mouth, which must be taken before winning the ko. Example:
Here is a simple three-stage ko.
In principle, there can be very long multi-stage kos, but it is rare to see one longer than three stages in an actual game.
See also Counting A Three Stage Ko.