A lot goes on during a simple click of the camera shutter. In single lens reflex (SLR) cameras, a mirror routes the light from the lens up through the viewfinder. This means that the photographer sees the same thing that the film (or sensor chip, in digital cameras) sees. It also means that the mirror has to get out of the way for the picture to be taken.
In the video below, you can see the mirror flip up in slow motion, the shutter open, and the mirror snap back into place.
Canon 5D shutter video by James Pearman.
The above video, taken at 2,000 frames per second, shows the focal-plane shutter actuating for a brief exposure, probably 1/25o of a second. You can tell the exposure is short because the entire image sensor is only revealed for a moment even in slow motion.
For shorter exposures, the shutter would not open beyond a slit. Instead, the first leaf of the shutter mechanism would be followed even more closely by the second. So the top and bottom of photos taken by a camera such as this one can actually capture separate moments in time. Pretty cool.
Here's a different camera (a film SLR) shown from the back, again filmed at 2,000 frames a second. You can see how the two planes of the shutter reposition themselves for the next shot while staying closed.