Just like shutters on a window, your camera has a shutter which opens and closes to allow a certain amount of light to hit the sensor. Your shutter is controlled by a time value and is simply the amount of time the sensor is “open” to the light. Obviously the longer you leave the shutters open, the more light gets into a room. Same thing with a camera. The shutter can be “dragged”, or left open for a long time. This leads to the aforementioned really cool time-based effects (e.g. car lights streaking, star trails, action sports motion blur). Conversely, the shutter must open only briefly to freeze a quick motion in time.
Old motion picture film cameras often used a rotary shutter: a simple half-circle that rotated to expose the film and then cover it as film advanced to the next frame. Because the film was running at 24 fps, and the “180º half-pie” rotary shutter covered it for half the time the film sat in place behind the gate, the approximate exposure time per frame was around 1/48 of a second.
So if you want the motion blur associated with what we’ve historically seen at the movies, 24fps and 1/48 of a second shutter is a great place to start. If you’re using a hybrid camera you’ll likely use 1/50 instead. Don’t sweat it. You’re friends will never see the difference and anyone who claims to is probably not a very healthy friend for you. Which brings to me to a broader point: don’t get fixated on this. If you’re shooting in low light, feel free to drop your shutter to 1/30 of a second, or if you need to reduce light a bit to avoid clipping please increase your shutter to 1/80 or so. Most people won’t see the difference, especially if it saves you from bigger exposure problems. When shooting at high frame rates, don’t freak out too much about doubling your shutter speed. You’ll start to lose a lot of light, and the amount of motion blur will be different than the “standard” anyway. When you start to play back at slower frame rates than you shoot at, sometimes the faster shutter speed is nice in that we have more time to notice the blur as an audience because the clip is playing slower, but again, do some tests and see what you like before trusting me or the internets.
Shutter speeds for still photography are a very subjective thing, and it boils down to deciding how much motion you want visible. If you want to show motion in the frame use a longer, slower, shutter speed. If you want movement to be precise and crisp, use a faster shutter speed. But it’s not just your subject that moves; the other extremely important consideration is how much visible camera motion you want. For most applications in still photography, you’re minimizing apparent camera motion because you want at least something in the frame sharp and if you move the camera then everything in the frame goes blurry. That is, unless your subject is also moving in the same direction as the camera, in which case the subject can remain relatively clear with the surrounding environment blurry.
The typical rule of thumb for a sharp, hand-held exposure was using your focal length to determine your shutter speed. The focal length number in millimeters becomes the denominator of the shutter fraction. The more zoomed in you are, the more visible any camera motion will be, just like when you’re looking through binoculars or a telescope and small movements get exaggerated. So if you’re shooting on a 400mm lens, use at least a 1/400 shutter speed. Truth told, modern cameras with stabilization both in lens and body are making it much easier to fudge this rule by several stops, but it at least gets you thinking about camera motion as it relates to shutter speed. Also, visit the focal length section if you need a refresher on determining your “effective focal length” and how that relates to sensor size. And, again, don’t forget to take into account subject motion. A kid frolicking in a playground isn’t going to look sharp at 1/50 just because you’re using a 50mm lens.
*Sony hybrid cameras, like the ones I frequently recommend here, exhibit a weird behavior where if you drop the shutter speed below the frame rate they start doubling frames.