The wagon wheel illusion that you see in old western movies: Even though the wagon is moving at a rapid rate, the wheel appears to move slowly, in the opposite direction, or not at all. This is because the frame rate of the film in the camera is too slow to get an accurate representation of the wheel's motion: If one spoke in the wheel points straight up every 1/24th of a second, then it will appear to the camera that the wheel is not in motion. If you have ever been to some sort of dance party that had a strobe light and an electric fan running, you've probably seen the same effect IRL.

Aliasing can also be a dangerous pitfall in data collection using digital elements. If my sampler takes 100 readings a second, then it will be able to accurately characterize a 5Hz sine wave, but a sine wave input of 105Hz will end up looking exactly the same to the instrument. I have been told that this sort of thing has crashed rockets in the past, when an unexpectedly high-frequency vibration has caused some feedback mechanism to erroneously compensate.

Moral of the story: Make sure that your bandwidth is plenty high for what you want to see, and stick a low-pass filter on the input to make sure that you don't see anything else. And keep Nyquist's Theorem in mind, too.