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“Rise” (up-and-down variations) and “shift” (side-to-side variations) are methods of using only a section of a lens’s “image circle” (the cone of light that shines from the subject, through the lens, and onto the recording surface, whether it is a digital sensor or a piece of film).
If you imagine a projector shining a beam of light onto a projection screen, the effects of rise and shift are akin to holding a piece of posterboard in front of various parts of the screen so that only part of the full view is seen on the posterboard.
Rise and shift are most often used in order to keep the camera level (for rise) or “square on” to the subject (for shift) so that square and rectangular objects in the scene (especially buildings) do not look wedge-shaped.
Rise and shift effects can be achieved with a “shift lens” (it is rotated 90 degrees to achieve “rise”), or with a “view camera,” or even by first taking a photo with a wide-angle lens on a camera that is held level and then cropping to the desired subject.
See also FAQ #810.