This is not strictly related only to digital cameras. Older film based cameras often feature a focal plane curtain shutter, which "rolls" across the film plane and exposes the film horizontally or vertically. A very famous example of this is the racing car picture taken in 1913 by Jacques Henri Lartigue using a 4x5 Speed Graphic camera.
The shutter moves relatively slowly on this camera, when compared with modern SLRs, which gives the distortion of the racing car. The distortion is especially visible in the wheels, which appear to be leaning forward. This effect was later copied by cartoonists when they wanted to give the impression of speed.
The term "rolling shutter" is not only used to describe the mechanical or electronic shutter implementation, but also the distortion itself. You're not likely to see this distortion effect when using Micro Four Thirds cameras, since the shutter is moving very fast. You must take a picture of something very fast moving to be able see the effect.
One such example is recording a rotating propeller. Here are two images captured from a 1080-line video recording using the Panasonic Lumix GH1 camera and the Panasonic Lumix 20mm f/1.7 pancake lens.
While the camera is exposing the video frame vertically, the propeller is moving, making it look bent. In reality, the propeller blades are straight, of course.
Rolling shutter distortion artefacts can be very annoying in video. You may see this as a "wobbling" when the photographer is panning horizontally. Luckily, the GH1 handles this rather good when recording videos. In normal use, you'll need to look very closely to find these artefacts in typical video captures.
I have made a study of the amount or rolling shutter distortion in Panasonic GH1 and GH2. It seems that the GH2 is slightly better than the GH1. Even though I generate rolling shutter artefacts in my study, they are not commonly seen in real life usage. So this is not any big issue at all.
The concept of rolling shutter becomes quite relevant again with the electronic shutter option available with cameras like the Lumix G5, G6, GH3, GM1, GX7. Using this electronic shutter feature can produce large rolling shutter effects. Using one of these cameras, you can again make pictures like the one from 1913:
Read more about how to do it here.
Comparison with other systems
The Nikon 1 mirrorless cameras were designed with electronic shutter in mind from the start. The Nikon 1 S1 10MP camera has a 1/60s readout, six times faster than that of the Panasonic GH3. The Nikon 1 J3 14MP camera is even better, with an electronic shutter capable of reading the whole image during 1/80S.
Only the Nikon 1 V2 camera features a mechanical shutter at all, usable when you want to be sure the image comes out without any rolling shutter artefacts. Further, the camera can take full 14MP images at a staggering 60FPS rate using the electronic shutter, opening up for very interesting uses.
From the Sony NEX line, some of the recent cameras, e.g., NEX-5R and NEX-6 have an electronic front curtain shutter. This means that the mechanical shutter is not closed prior to the start of the exposure. The only mechanical shutter is the one which closes when the exposure is stopped, the rear curtain. The advantage is obvious: It reduces the vibrations before the exposure commences, and also reduces the number of mechanical movements overall, while still avoiding the rolling shutter artefacts discussed above.