<eddit> Due to this working malarky, you've posted twice, and HK & Tim once in the time it's taken to compose this. Sorry. Hope it's still remotely useful. </eddit>
In really simple terms, to achieve a given exposure, you have three variables; Exposure time, Aperture & Sensitivity (ISO). Each has a disadvantage to being increased too much. (I fear a lot of what I talk about here will push beyond P&Ss in places, and into the realms of D-SLRs, but if you see the differences between the two types of camera you'll better understand your own).
Increasing exposure time introduces the potential for movement, in the form of both camera shake and motion blurr (the latter can sometimes be desirable - think moving water). Camera shake can be eliminated by using a tripod, beanbag, or wall and a remote release (or setting the timer on the camera) to minimise operator contact with the camera as the shot is fired. If you're really anal about this you can use mirror lock-up too on a D-SLR (essentially minimising the amount of internal moving parts before the shutter moves).
The aperture is the size of the hole in the lens. A bigger hole = more light, which increases the exposure. A big aperture is given a small f-number (f/2.8, say), and a large aperture is given a small f-number (f/22). This is because the aperture diameter is expressed as the focal length over the f-number (eg. a 100mm f/4 lens will have an aperture of 25mm, a 200mm f/4 lens will have a 50mm aperture). Increasing aperture (decreasing f-number) decreases the depth of field. That is, the depth of the region that is in focus. For some good examples of this, see
Heather's Article.. Further disadvantages with large aperture lenses are on the wallet (big glass = pricey glass), and the back, as big lenses are heavy and not so discreet.
Finally ISO. In film terms this was the sensitivity of the film, in digital it's the sensor. That is, how quickly it reacts to a given amount of light. With film you could get graining with very sensitive films (High sensitivity required bigger crystals, which show up more), and with digital you get noise (due to increased voltages/currents/whatnot). Whilst B&W film grain could be attractive if controlled, digital noise rarely is. Newer cameras are getting better at controlling it, as are those with bigger sensors (lower pixel density = less chance of noise, but then lower MP output files). Typically aim to shoot as low a ISO as you can.
A stop (or f-stop) is, if you will, the unit of exposure. Increasing the aperture by one stop (eg. f/4 -> f/2.8) doubles the amount of light received. You can therefore, halve the exposure time, to maintain the same exposure, or you could be trying to
increase the exposure by one stop. And to include ISO, if you've increased the aperture by one stop, and kept the same exposure time, you'll want to halve the sensitivity, so, for example, move from ISO 800 to ISO 400.