Going back to our tub analogy where how open the faucet is corresponds to the aperture, and the depth of water we want in our tub equates to the ISO, the simplest aspect of our exposure triangle to get our heads around is shutter speed, or how long we leave the faucet open.

So when would we prioritize shutter speed by setting a speed and letting the camera decide the aperture? The key use is when you have a subject in motion and either to need a set shutter speed to prevent motion blur or a speed to allow a certain amount of subject blur.

One example would be one of our regular subjects, birds in flight. If you know it takes 1/800 second to freeze the tips of the wings when taking pictures of a pelican then you would set that as your shutter speed and let the camera take care of the rest. Or if the subject is soccer players, and you know a running person's foot has a small amount of blur at 1/300 second then go for that speed.  If you don't know what shutter speed you need to get the freeze you want then take a few test pictures at different shutter speeds and zoom in on the review screen on the back of your camera.

Conversely if you want subject motion, like at a fairground at night or moving traffic for light trails,  or to show flowing motion of a wedding reception first dance then choose slower speeds.  I'll use several seconds for fairground lights to over one minute for traffic trails, to 1/4th to 1/30th second at wedding receptions.

Other scenarios where I would use shutter speed priority would be an airshow where there are propellor driven planes to make sure I get a good amount of blur in the propellor (the 'rule' is a full circular blur).  Another is taking images of moving cars or motorcycles where, if the shutter speed is too high, then the background and the wheels are so sharp it looks like the subjects are stationary. 

The main point is to be deliberate, go out with the intention of taking a specific type of image and take control of your camera to make it deliver those images.