I am sure this is a common question, but to find it easily, what is significant about 1/125 and 1/30? My teacher asked us and I have no idea. I can not find it anywhere. Why are these shutter speeds so important?
I understand that you mention those two specific speeds for some reason, and not as a general example.
My best guess on why your teacher is asking is:
Traditionally 1/125 was the maximum shutter speed to sync a flash, because higher velocities were achieved by starting to close the shutter before it has finished "opening", so a flash would only expose a fraction of the frame.
Now a common velocity for this is 1/180 - 1/250.
The 1/30 is probably considered the minimum velocity on which a photographer can hold a camera by hand and not take a shaky photo. This minimum speed depends on the photographer's skill, the focal length, the sensor size, and whether the camera or lens has a stabilizer integrated.
This 1/30 probably refers to a 50mm equivalent lens.
The amount of light hitting the light-sensitive material (film, digital sensor, collodion covered plate, etc) is measured in stops. Essentially, this is your exposure. One of the ways to control the amount of light reaching your light-sensitive material is with the speed of the shutter. Shutter speed is measured in seconds and fractions there of. The difference between your 1/125 (of a second) and 1/30 (of a second) is two full stops. With each stop you are doubling the amount of light, so with the shutter speeds you have given you will effectively quadruple the amount of light being let onto the material assuming you haven't changed your aperture (fstop), which is another way to change the amount of light coming in to the camera. Now, that being said, no shutter speed is any more or any less important than any other shutter speed. Some you will use more than others, but each one is important if it will give you the correct exposure for any particular shot.