All photography is based on a reaction of a receptor (a film or digitalized sensors) to the presence of light.
Exposure is simply the amount of light you let come in to the receptors.
There are several ways to control exposure, and one can also have some play by using receptors that have different levels of sensitivity (different film speeds).
Shutter speed is the principal control. The amount of time that the camera is open to the light is what most people consider exposure. Thus, in your example, a long exposure time is desired because the amount of light coming in per unit time is very low.
F-stop (aperture control) is another way to control exposure. If the iris (the opening in the camera) is opened wide, the amount of light coming in per unit time will be a lot higher than if it is closed to a pinpoint.
One can get some interesting effects by playing with shutter speed, aperture, lens filters, and differing film speeds. It is this interplay of the various things that makes photography interesting and fun.