Using a slow shutter for effect can dramatically change your photo.
A moving car with blurred wheels gives the feeling of movement.
A person moving through a room shows life and feeling. Coffee beans roasting, showing them turning - I can nearly smell the beans by just looking at the image. 

With the coffee bean image above I used a slow shutter, probably something like 1/30 or 1/15. How slow you go depends on the subject you are photographing. Playing around is the best way of learning. Experiment with the effects. 

1/125 = one twenty-fifth of a second
1/30 = one thirtieth of a second
1/5 = one fifth of a second
you get the idea...

If you are shooting with an iPhone, there is an App called Slow Shutter (and various others) or Manual which allows you complete freedom and control over settings when taking a pic. 

The faster the object you are photographing, the faster the shutter speed you can have. All depending on the effect you want. 
I have done photos in the studio with the shutter open for seconds and moved people around. 
I have photographed cars and movement on the street at 1/60. 
I have photographed details of dancers on stage at 1/30. 

There is no right or wrong shutter speed - it all depends on what you want. Experiment!