Shutter speed is what determines how long your sensor/film is exposed to light. The less time it is open the less chance for blur in the image, whatever the source might be, but also less light. Because of this the faster your subject is moving the faster your shutter generally needs to be, and the more light you will need on that subject. This is true whether its a car race, runners, or dancers at a wedding. Even with ISO and Aperture set to their maximum values only so much light might be available, forcing either a slower shutter or some kind of artificial lighting be added to the environment. If a slower shutter is not an option then lighting is required.
Recall the ISO explanation, the higher the ISO value the more sensitive the sensor/film is to light, but also it gets more and more degraded, either in terms of grain with film, or static with a sensor. (No, thats not grain you see, its just static, very similar to a weak radio station)
So, shutter speed. This controls the time of the exposure. If shooting a moving car you want some blur or the car looks parked, so wheel blur is good. If trying to freeze motion, like a splash of water, a very fast shutter is needed. If shooting a waterfall, again some blur is usually desirable. So, the shutter, besides being a technical aspect of the camera operation, it is also part of the artistic control you have access to.
One other note on your shutter thoughts… If you are using a flash you need to be aware of sync speed. In most cases this is going to restrict you to 1/200th or 1/250th of a second. Sync speed will be discussed in detail later, but it is the point at which your given camera no longer fully exposes the sensor. In modern cameras there is no single shutter, it is comprised of 2 curtains. The first one moves across creating an opening. Below sync speed and the sensor will be fully exposed to the light until the second curtain comes across, closing the shutter. Above sync speed and that second curtain starts across before the first shutter completes its trip. In these situations flash units that support it will fire multiple times in such a way that the full span of the sensor will be uniformly lit. The downside is that this mode uses a reduced light power, so you do not get the full power of your flash. In general, where possible, stay below sync speed and you will be good to go, though there are games to be played for certain effects where you tell the flash to fire late, but thats another article.