For different usage, bigger aperture (smaller number) means more light pass through onto the film, or the sensor, which allows you to take photos of poorer light condition. Besides contributing to the exposure of the film or sensor, the aperture also controls depth of field .
Depth of field basically refers to the distance in front of and beyond the sharply focused subject of the picture. For example, with a standard lens using it's widest aperture, let's say f1.8, and a subject of 1.8m from the camera, very little of the background would be in focus. But, if the aperture is set to f16 or even f22, more of the background and foreground will be sharp, which is ideal for group shots or landscape etc.
Lens with bigger aperture will normally cost much more.