Pushing film or pulling film is using an ASA or ISO that is different from what the film is intended. Using ASA 100 film and setting your camera to ASA 200 or ASA 400 is pushing your film. Going the other direction using ASA 200 and processing it as if it was ASA 100 is pulling film, not done as often as pushing. When it is developed you must inform the lab you have pushed the film or pulled the film and by how much so they can adjust their development process. For pushing, it would be left in the developer longer or use an slightly different chemical developer. The end result of pushing the film would be an image that has more contrast than it normally would. There is not an exact comparison to digital but one example of using a similar process of pushing with digital would be when you need a higher ISO than your camera is capable, you can underexpose the image and adjust it during editing. The difference is the lack of consistency in the contrast digitally which results in more digital noise than grain. Instead try using an editing program that gives you control in adding grain. One such program is Nik Software's, Efex Pro.
Dragging the Shutter**
Dragging the shutter is a process of using a slower shutter speed when combining ambient light with flash or strobe light. This is important when photographing people in low light in front of something you also want to see in the photograph. Your flash will light up your subjects but often when you shutter speed is set at or near the synch speed you will get a dark background despite the low light in the scene. You can use a slower shutter speed to allow ambient light into the scene while the flash will freeze your subject and keep them in focus despite the slower shutter speed. This isn't always perfect and can introduce ghosting into your image but it can be effective in showing the scene and the subject. With a point and shoot try the night setting for a similar approach.