There are many factors that play into the minimum shutter speed, but the simple rule of thumb is that you need to be faster than your effective focal length if you're holding the camera. For example, if you're using the equivalent of a 50mm lens, your shutter speed needs to be faster than 1/50 sec to get a crisp photo, and if you're using the equivalent of a 200mm lens, your shutter speed needs to be faster than 1/200 sec, etc
Of course, there are different scenarios that alter this. For starters, notice that I said the 'equivalent' of a 200mm lens. This is because most camera sensors are smaller than 35mm, and because they're smaller, the light angles are altered as they come in from the lens to the sensor. Because of this, most DSLR cameras have an effective focal length of 1.5 times what it says on the lens; for example, a 50mm lens acts like a 75mm lens. This isn't always the case, however, so it's good to check your camera's documentation. The high end cameras sometimes have a larger sensor, which brings the effective focal length back to what it says on the lens.
Another thing to keep in mind is that some lenses have Image Stabilization or Vibration Reduction. Both of these terms mean the same, but Canon uses the term Image Stabilization or IS and Nikon uses Vibration Reduction or VR. What this means is that the lens has a way to counter the effects of camera shake. Not all lenses are created equal. Some have better VR or IS than others so you'll want to check the documentation and experiment with the lenses; however, it's common to be able to slow down the shutter speed by 2-4 or more stops. This is very important for telephoto lenses because they're a lot harder to keep steady and thus require a lot faster shutter speed. To put it into perspective, it's pretty easy to have enough light at 1/50th of a second but a lot harder at 1/800.
Monopods reduce the effects of camera shake and thus the shutter speed can be slowed down by several stops. The camera still moves slightly, however, because it's still braced by a human. Of course, tripods virtually remove the effects of camera shake except for the moment the shutter release button is pressed. In some cases, such as long exposure shots, it's necessary to use a tripod and a remote shutter because pressing the button actually moves the camera slightly.