There are three main adjustments you can make to a camera: The ISO, Aperture, and Shutter Speed. I've explained both ISO and Aperture before, so it only makes sense to explain what exactly Shutter Speed is and how to use it to your advantage.
In short, Shutter Speed is the amount of time your camera's shutter is open. Sounds simple, right? Well, Shutter Speed can get a lot of more complicated depending on the situation.
Every time you’re setting up to take a shot think about exactly what you want the depth of field to look like and set your aperture accordingly (explained here). Once you have your aperture set, ask yourself what you want the final shot to look like.
If you want to freeze something that’s moving fast like a car/person in motion, a bird’s wings, etc., you’ll want to have a fast shutter speed. How fast is really up to you. If you want to capture a humming bird’s wings, you’ll have to set your camera up to at least 1/1000 of a second. However, you might be happier with a little blur in the wings and make the shutter speed a bit slower.
So when do you use slow shutter speeds? Typically there are two scenarios for slow shutter speeds: low light / night time photos and creative photos. For either situation, a tripod is your best friend. You’ll want to use slower shutter speeds to allow light to hit your sensor over a longer period of time which will make dimly lit areas light up beyond what your eyes can see.
Low light / night time photos are an obvious one. You don’t have enough light to use a fast shutter speed so the only two options are to use a slower shutter speed or increase your ISO which will increase the noise in your photograph (explained here). This is also true when using ND filters which makes the scene "X" stops darker.
When I say creative shots, I mean scenarios in which you'll want to use a slow shutter speed to do things like paint with light, smoothing out water/clouds, taking pictures of stars, clearing moving people from a shot, etc. A sensor captures information as long as the shutter is open so anything that moves will blur/smooth over time. If you want to do any of the above scenarios during the day or in a very well lit place, attach an ND filter to your lens and you’ll be good to go.
For a fun project, try capturing a busy street with a long shutter speed. If every person moved during the entire time the shutter was open they will disappear in the shot. You can make places look empty or abandoned that way.
There is, of course, one question that comes up a lot. What do I set my shutter speed to if I don’t have my tripod and the place is dimly lit? The best practice is to set your shutter speed to at least 1/your-focal-length (ex. 50mm = 1/50s). That’s not set in stone, but it’s a great starting point for most handheld shots. DSLR and even point-and-shoots are handling ISOs better than ever. Don’t be afraid to increase it a little bit, but remember the lower the better for more situations. I typically try and keep mine at 100.
If you have any further questions or I didn’t explain something well enough, please let me know below. Otherwise good luck, and happy shooting!