A diopter is a filter that works similar to a magnifying glass, making a lens focus closer. A split diopter has it on only one side, meaning that one side focuses close and the other farther away. This allows objects at different distances from the camera (usually on the left and right side of the screen) to be in focus simultaneously.
Used to great effect in The Thing (1982), for example, where dark sets disguised the blurred region in the middle of the split shot. Sometimes used by first-time directors as a way to "show off" their technical skills.
Although it can be, and often is, used as a special effect, a scene as simple as two characters conversing might call for a split diopter (or a tilt/shift lens, which provides a similar effect by allowing the focal plane to be tilted so it is no longer parallel to the film) if the characters are not at the same distance from the camera. Of course, many shots are staged with the characters equally distant from the camera precisely because no special optics are needed.