Usually, one would want to push process film that was underexposed. For overexposed film, one would need to pull the processing to get 'proper' exposure.
Let's talk about push and pull for a moment.
When shooting film, if one exposes for a film speed faster than the actual speed of the film, one is technically pulling (underexposing) the film. The film then needs to be pushed in the other direction (over developed) when processed. But many photographers say they are pushing their film while underexposing during shooting when what they actually mean is that they are pulling (underexposing) their film when shooting because they intend to push (overdevelop) the film when processing.
On the other hand, overexposing the film when shooting it is pushing the exposure brighter. This needs to be counteracted by pulling the development time back to compensate.
Like many things in photography, it's about ratios and reciprocals. If I expose by giving the film three-halves (3/2) as much light as I should, I then need to develop the film to two-thirds (2/3) the typical development. If I expose the film to one-half (1/2) the 'normal' amount of light needed, I then need to develop it to twice (2/1) the 'normal' development. (Please note that development times are not linear - twice the development does not necessarily mean developing for exactly twice as long).
For instance, if I have ISO 100 speed film in the camera and choose to expose for ISO 400 speed film, I'm underexposing by two stops. It is true that ISO 400 film is more sensitive (and thus brighter) than ISO 100 film. But when I change my camera's ISO setting to '400' I'm not actually changing the film's sensitivity to ISO 400 - I've still got ISO 100 film loaded! What I am doing is reducing the amount of light the camera's meter says my film needs to be 'properly' exposed. So my film winds up underexposed by two stops.
When that film is developed, it needs to be pushed two stops brighter to get more or less 'proper' exposure. But the result will be lower contrast (brighter shadows) and coarser grain than if the film had been metered and exposed for ISO 100 and normally developed.
If you overexpose your film, you would normally pull process it by developing it for less time than normal. This will tend to increase contrast in the mid-tones and dark areas and muddy the shadows, but highlights that have been totally blown will not show detail, they'll just appear to be a uniform bright gray when printed.
If you push your film one stop when exposing (give it twice as much light as the meter calls for) and then also push your film in development (develop for longer than normal), you'll get a negative with very high density that will translate to prints (or scans) with a low contrast but washed out look with blown highlights if there were any in the scene and no true black areas.