I've read a pretty cool, although somewhat useless, example of this a while ago. Imagine towers with powerful projectors pointing up spaced out 10 miles apart all the way around equator. You configure towers such that they sequentially activate projectors for a small fraction of a second, and only one tower is active at any given time, and at least one tower is active at any given point of time, and towers must activate sequentially.

Now with this setup if you dial down activation time so that it is smaller then 53.7 microseconds [0] next tower will not be able to receive any communication from previous tower on when to activate, signal would just not travel fast enough. Activations of towers would need to be scheduled. Now consider an observer looking at Earth with a powerful telescope from a significant distance. To him there would be an object emitting light that is moving around earth faster than light.

[0] 10(miles towers apart) / 186282(light speed in miles/second in vacuum)= ~0.0000537 seconds = 53.7 microseconds

It's when you (for example) sweep a beam across a field. The spot of light can move really quickly, even crossing the field faster than light itself could, but it's not conveying information from where it's been to where it's going. Nothing physical is moving at anything even approaching the speed of light.

Typical example is a wave crashing on the shore at a slight angle. Wave may be going only a few mph. But the wavefront breaking on the sand will appear to move horizontally at tremendous speed.