If your photography teacher was telling you that the exposure mode has a "huge impact" on photographs, he deserves to be fired and then have all of his equipment confiscated and burned.
You have four main modes on the camera, M, Av, Tv, and P. They each allow you to control the aperture and shutter speed, but they do it a little differently. You can (and should) get the same results regardless of which mode you use. The mode you choose is impacted by what you are comfortable with and what the scene is.
M- Manual. It allows you to set the shutter and aperture independently, and without relying on the camera's meter.
Av or A- Aperture Priority mode. You set the aperture, and the camera will pick a shutter speed based on the meter. You can still adjust shutter speed with exposure compensation.
Tv or S- Shutter Priority Mode. Works the same as Av, but you set the shutter speed, and the camera used the meter for the aperture.
P- Program mode. The camera picks a shutter speed and aperture, and then allow you to scroll through a series of equivalent exposures. You can adjust bias using the exposure compensation.
What you use might depend on exactly what you are doing. It is a total myth that one mode gives any more "control" than any other. A decent photographer will already have a shutter speed and aperture in their head, and can easily get there using any of the above modes.
The choice of mode, just depends on efficiency. For example, for someone whose main concern is controlling depth of field, they may use Av mode because they want a specific aperture, and will let the shutter speed fall where ever it does. If someone is trying to balance ambient light and off camera flash, they may use M mode since they are using the shutter speed for ambient light and the aperture for flash.
Edit:
As to what is used more, it depends on the photographer. As a wedding photographer, I am probably in Av mode 90% of the time, and the rest of the time I shoot in manual. I don't think I ever use Tv. A sports photographer who needs to freeze motion, and therefore requires a minimum shutter speed might use Tv mode because the shutter speed is more important than the depth of field.