Film emulsions are based on silver halides, whose nature it is to create a dark areas wherever exposed to light. (This is explained further in the Film entry.)
Since brightness in the scene creates darkness on the film (and vice versa), the natural outcome is a negative image. Humans cannot interpret this very well, and a negative is simply used as an intermediate stage in producing a print or a digital scan.
But there are cases where it's desirable to convert the exposed film itself into a positive image—e.g. to project a movie or a slide at a large size. Through a development process slightly modified from the the normal one, this is possible.
A first developer creates the normal negative image. But instead of next dissolving the unexposed silver halides with a fixer solution, instead a chemical is used to bleach away the negative silver image. The remaining silver halide grains left behind represent dark areas that the scene did not expose. By using a fogging developer (or flashing with light and then developing) these areas develop as dark, and a positive image results.
Not every film responds well to this treatment: Sufficient silver halides must remain after to form an image with a good range of tones. Also, the clear support of the film may lend an objectionable color tint.
Black & White Examples
One of the few B&W films specifically designed for reversal processing is Agfa Scala. Although it is now out of production, some stocks remain (as of early 2011). At one time Kodak offered a reversal-developing kit matched to its (discontinued) Panatomic-X film. Also, there is a movie-film version of Tri-X, 7266, specifically designed for reversal processing. The lab dr5.com is a specialist in reversal processing services; their website notes the suitability for this of many different film types.
For most photographers, B&W reversal processing is just a historical curiosity. However all color slide films are designed from the ground up for reversal development. Reversal is an essential step in E-6 processing, and for this reason slide films are sometimes termed "color reversal" films.
While color films are engineered to behave optimally in their intended process (be it E-6 or C-41), slide films will develop as negatives in process C-41; and print films will develop as positives in process E-6. This results in abnormal color palettes, but such cross-processing can be an interesting visual effect in some cases.