A short shutter-speed is needed to freeze an object in motion. The faster the object, the shorter the shutter-speed you need. It also depends if the motion is close or far and its direction, so we cannot tell you exactly how fast a shutter-speed is needed.

When buying a camera, you can look for the maximum shutter-speed of the camera. That lets you know how fast it can exposure. Usually, 1/2000s is very common and would freeze pretty much any person but if you are trying to freeze a racing car for example, it wont do. A few recent cameras go to 1/16,000s. Those are the Nikon 1 series of mirrorless cameras. Casio makes some fixed-lens that reach 1/40,000s.

There are object such as bullets which are way too fast for most cameras to capture and for that a fast shutter-speed is simulated by using a very short burst of flash. For that you need a camera with dedicated flash unit and controlled environment, so that the flash produces most of the illumination. Search of high-speed photography if that interests you.

As per the comments below, @dpollitt has a point. The object you are shooting must be in focus and autofocus can along focus so fast. Some cameras are better than others but there is a point when no autofocus can keep up with a speedy object. My advice is to use manual focus for objects which have a predictable path and set your focus to the place when the object will be when you take your shot.