
Shutter speed is arguably what is going to effect your photo the most. What it does can be inferred by it's name. Think about the basics of a digital camera. The exposed sensor takes the picture that is let in through the lens.
That being said: The shutter blocks the image sensor from the light. The speed is how long the sensor is exposed to that light, which creates the picture.
The flow of the light is augmented by the current aperture of the lens, which will be discussed in another post.
As you can see from the series images on the left, courtesy of Wikipedia, the shutter speed dictates quite a few things. This camera is clearly held stable by a tripod, and is taking pictures of the same spot. Each spot is take under a different shutter speed, all the way from 1/800th of a second to a single second. As you may know, motion with the shutter open for too long is going to blur your image. This blur can be shown somewhat tastefully to show motion, but the extreme will render a picture abstract and meaningless. The water looks very fluid and slick with the long shutter speed, the water is blurred but in a path so that it's atheistically pleasing. It's important to note that if the camera wasn't on a tripod, this would not have been achieved. Alternatively, you see the jagged edges of drops of water being caught in mid air with the fastest of the shutter speeds, which is also makes an attractive image. This kind of picture however, could be caught without a tripod mind you - whereas the shutter speed was fast enough so that a steady hand wouldn't ruin the photo.
Exposure also refers to, more commonly, how bright or dark the image is. If the shutter is open for too long, your image is going to be too bright and you wont be able to see anything. Even slightly over exposed pictures have details in highlights which are blown out.
If your shutter is too quick- you risk the opposite. Your picture will be too dark, and the slightest of shadows will seem like the darkest abyss.
These are two things you must balance, along side keeping in mind how you want motion to effect your photo. It makes for difficult trade offs in situations, with the most prominent that comes to mind being low light photography. How would one capture a crisp image without blurring in low light conditions? One would thing that the shutter would just need to be open for longer to compensate, but keep in mind it would be blurred.
This is where other elements of exposure come in to play, and can help you control other elements of your photo including light, but also detail and focus.