How does a hovercraft fly?

They technically do both, as they fly when moving (nothing needs to touch the ground) and hover when standing still (unless they are turned off). This is assuming, of course, that altitudes less than a foot are considered flying.

To add: hovercraft can lift off, without their skirt, as they make fond use of the so-called "ground effect", much like helicopters do at a greater altitude. They keep lifted by sitting on the rotation of air beneath the vehicle and move forward by propelling the air behind the vehicle, rather than sitting on a big, inflated balloon (like many people tend to think). The skirt only serves as a mean for stability.

Technically, the hovercraft does fly, but is not considered to be an aircraft, whereas it does not elevate to great heights. Thus it is called a "hovercraft". People who operate hovercrafts, however, are generally referred to as pilots.