A false-color image is an image that depicts a subject in colors that differ from human perception of the same subject.
A true-color image of a subject is an image that appears to the human eye just like the original subject would: a green tree appears green in the image, a red apple red, a blue sky blue, etc. When applied to black-and-white images, true-color means that the perceived lightness of a subject is preserved in its depiction.
In a false-color image this close correspondence between subject color and image color is violated. That can happen in many ways. For example, a photographic negative could be called a false-color image, since it shows the complementary colors of its subject. However, the term "false-color" is typically used to describe images whose colors represent measured intensities outside the visible portion of the electromagnetic spectrum (or outside the electromagnetic spectrum altogether).
For example infrared film captures a small region of the near infrared spectrum (wavelengths from about 700 nm to 900 nm). Black-and-white infrared film maps the average intensity in this portion of the spectrum to a "false" gray value. Typical color infrared film is sensitive in the visible and near infrared range; it is usually used, in conjunction with a blue-cut filter, for shifting the near infrared spectrum into the visible spectrum (ca. 400 nm to 700 nm). Loosely speaking, this process effectively subtracts ca. 150 nm from the subject wavelength: a green subject is depicted as blue; a red subject is depicted as green; and an infrared subject is depicted as red (blue subjects cannot be depicted; they appear black).
More generally, false-color images can be generated by digital image processing techniques to represent up to three independent measurements over a two-dimensional map or image. For example, a satellite might capture average intensities in short ranges of the ultraviolet spectrum and the near infrared spectrum. These measurements can be combined with altitude information in such a way that blue represents ultraviolet, green represents altitude, and red represents infrared. Then a bright yellow pixel indicates a high-altitude object that emits or reflects infrared light but not ultraviolet light; a magenta pixel indicates a low-altitude object that emits or reflects both infrared and ultraviolet light; etc. Because human visual perception is limited to three independent coordinates, at most three measurements can be depicted in such a way.
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