Discovered in the 1860s by Ernst Mach (hence the name), Mach Bands are actually a set of interrelated phenomena. Take a look at this image:
The individual bands should appear as gradients, and they may even appear to be curved. In fact, they are all solid colors. Now look at this one:
If you look closely at the area above the center two arrows, you should see a thin bright line (left-middle arrow) and a thin dark line (right-middle arrow). Once again, this is despite the fact that each of the three areas (dark, light, and in between) are solid colors.
This figure (from here) will aid in explaining these phenomena:
In the graph on the bottom, the black line represents the actual luminance of the figure. The red line reflects the perceived luminance. The red line's deviation from the black line represents the Mach Band phenomena, and the little spikes over and above the black line represent the bright and dark lines you see in the second figure above.
Why does this occur? The traditional, and still generally accepted explanation goes something like this. Light reflecting from each of the colored bands in the figures above will strike many different center-surround receptive fields (represented by the circles with + and - signs in the figure above the graph). Some of these receptive fields will actually overlap the boundaries of the bands. This causes these fields to be excited at different rates than if they received light only from one of the bands. These differing firing rates result in the fields sending signals that cause the visual system to believe slightly different luminances are present, resulting in the gradient, and at the edges, the bright or dark lines.
However, this explanation has been challenged recently by Dale Purves and his colleagues1. They argue that Mach Bands are actually the result of the properties of reflected light and the frequency with which we encounter curved surfaces in the environment. Because of the properties of reflected light, curved sufraces typically exhibit what Purves and his colleagues call "highlights" and "lowlights," which are essentially the same as Mach Bands. They argue that Mach Band phenomena therefore result from the visual system being tricked into thinking it's looking at a curved surface. To demonstrate this, they have shown that Mach Bands are stronger when the visual context is such that the illusion of a curved surface is stronger, and they are weaker when the context makes it clear that the image is flat. They argue that the traditional receptive-field explanation can't account for these results, and that instead the results indicate that the visual system has adapted to perceive "highlights" and "lowlights" on curved surfaces, and Mach Band-producing stimuli like those above activate this property of our visual system.
1Lotto, R.B., Williams, S.M., & Purves, D. (1999). Mach bands as empirically derived associations. Proceedings of the National Academy of Sciences, 96, 5245-5250.