Question about pushing film

I'm just a newbie who just started shooting film and I came across a blog where the guy shot Ektar100 rated at 800 and pushed 3 stops. How does the images differ from the same film shot at 100?
That led me to wonder how other films would look with similar method. I know wedding photographers shoot portrait at 2-3 stops overexposed to get that look. So what if you rate Potra400 at 1600 and push 3-4 stops? How would the images differ if the same film was rated and shot at 200 or 100? Does this method work better with certain films like Ektar?
I guess my question is how does pushing film affect image quality and color, etc.

If you rate Ektar at 800 you are under-exposing it by 3 stops, not over-exposing it.
As with any negative film, if you under-expose film by 3 stops, you lose shadow and near shadow detail - quite a bit of it. With normal processing, that would give you empty shadows and lower contrast mid-tones. Only the highlights will look fairly pleasing.
If you push process, you won't gain any meaningful shadow density, but it will improve the contrast of the mid-tones. It may result in the highlights becoming very contrasty, and losing some detail.

You do not really increase speed, you increase contrast. This increases grain and loses overall shadow to highlight detail. But, it gives the appearance of speed.

PE

Thanks for the explanation. So is this method mainly used when you need the shutter speed and/or you don't care too much about shadow detail and more concerned with mid-tones and highlights?

Pushing is surrounded with lots of myth. It is IMO applied mostly in the manner wishful thinking is applied.

Use the film closest to the speed you need, shoot at the speed you need to get the shot.

Shooting at the film speed closest to the speed you need is not always possible, sometimes you need to "invent" some film speed. Fortunately with C-41 films and B&W films, this is easy at the present time. You've got C-41 films ranging from ISO 100 all the way to 800, and B&W films are available from ISO 20 all the way up to around 1000 (Delta 3200 is a low-contrast film who's ISO according to Ilford is around 1000 but pushes well to 3200 with huge grain in my experience, at least in 35mm). My kitchen with the lights on is around EV 5, my living room with the lights on is around EV 4. I shot Portra 800 in both lighting conditions with my TLR's aperture and shutter speed set at f/3.5 and 1/25 sec, respectively (f/3.5 on this camera is wide open). I was underexposing by roughly 1/2 stop in the kitchen and 1 1/2 stops in the living room, and got usable, pleasing results.

With slide film (process E-6), this is not currently the case. We currently have ISO 50 and ISO 100 films available to us via current manufacture, and that's it! If you want a faster film, you have to "invent" the film speed, and this can be done to some degree by pushing. Comments by PE and Matt King apply here, however you can still get usable results if you're shooting E-6 indoors and pushing. I've shot Provia 100F indoors and out at EI 320 and had the lab push 2 stops. The indoor shots looked pretty good; the outdoor ones did see a contrast boost and did lose some highlight detail, as PE and Matt pointed out.

Negative films (C-41 and B&W) have lower contrast, so they do not gain contrast (in general, YMMV with some films) as quickly as slide film which is already contrasty to begin with. Therefore it seems that they can be pushed, but I'd say if you're looking to shoot available darkness with a color negative film, just grab the Portra 800 and don't push unless you're going up past EI3200. If you can't get Portra 800, then grab the 400 and push a stop or two if you're going to be shooting at EI's above 1600. I've also shot HP5+ at EI 1600 with a 2 stop push and the results came out fine, with finer grain than Delta 3200.