Exposure time is the time that the sensor/film of a camera is exposed to incident light which already goes through the lense. The exposure time can also be calculate by the shuttle speed (see this article).
In practice, the exposure time is used either to capture a moment (as in the first example below), or to stress on the motion (as in the second example).
In the image right below of short exposure time, the sensor was exposed on second; during which water in the fountain did not have time to move a very far, hence the “moment” was captured nicely (each droplets can bee seen individually). An indirect effect is that one can stilll see the some detail of the building behind. This “short exposure time” is used heavily in sport (race, football, etc.), in street photography, or even in very sunny day.
On the other hand, in the second image of long exposure time below, with the exposure time was of second. During that time, a water drop traveled quite a distance, so that the fountain became completely opaque, had more reflection and gave an shape-like impression, instead of a gathering of droplets as in the picture above. This effect is used almost all the time in any picture of falls (Niagara fall for example), or in night photography.