A peristaltic pump is any sort of pump that uses peristaltic movement to transfer liquid from one location to another. The easiest way to explain what peristaltic movement is, is to think of the intestines, colon, and esophagus. All three of these organs use peristaltic movement to move liquid and solids through the body.
The following definition is summarized from the online Encyclopedia Britannica: Peristalsis is the movement of long and circular muscles which occurs in wave-like contractions.
In dosing pumps, peristaltic movement is achieved by the use of rollers. These rollers pinch flexible tubing, thus forcing liquid through that tubing.