Most cameras these days have an adjustable ISO setting. The numbers relate directly back (like it was all that long ago) to the days of film, when the same numbers related to how quickly the film reacted to light. The lower the number, the slower the film was. Typically, slower films also gave much finer photographs. Most of the film typically found was rated at 100, 400, and 800; although there were many others available at higher end photography outlets (I actually still have a roll of Ilford 50 in the freezer I purchased for a shoot that never happened). The differences were both in the film carrier itself, as well as changes in the chemistry of the light sensitive layer. Back then, you had to have an idea of where you were going to be shooting, and load film accordingly. Bright, sunny day? load up a roll of 100. Family gathering at grandma's? Load up 400 (or sometimes 800, depending on how dark grandma's house is).
Since you can always shoot at smaller apertures, or faster shutter speeds to reduce the amount of light getting to the film, you can always just have 800 in camera (or now, have it set for 800), right? There's the rub. Yes, technically you could, but there are a couple drawbacks. First off, decent high speed film got expensive. Second, due to the difference in chemistry, it was also much grainier. The following photo is actually a digital photograph that I ran through a graphic editing filter that does a particularly good job of simulating different films.
The left half is simulating Kodak TMax 100 speed, which will be familiar to anyone who took a photography class. The right half is of Kodak TMax P3200, which was the fastest film you could typically find at the camera store. The difference could be used for the effect alone, I frequently did it myself on many occasions, depending on what I was shooting. Many times, though, the grainy effect would not benefit the final image.
The same type of thing happens in current digital cameras, due to the fact that as you turn up the sensitivity of the sensor, you're trying to get the same amount of information from less input. An audio geek or musician would refer to the outcome by discussing signal to noise ratio. Unfortunately, the effect in digital cameras is less photogenic than in film. The next photo shows exactly the same image taken in my digital SLR at two different ISO settings... the left is at 200, which is the lowest native setting I have available; the right is at 3200, the highest setting.
At first glance, it's really not to bad (it was actually lit reasonably nicely, so there was plenty of information for the camera to process). If zoomed in, though, the noise becomes more prevalent.
I feel compelled at this point to mention that I did no manipulation to the above photographs, other than zooming in on the chess board.
This is one of the few occasions that having a higher end camera comes in handy. For the most part, as I'm sure I've mentioned before, a camera is a black box with a controllable hole in it. Higher end cameras, however, do have much better electronics. The above photos were taken with my Nikon D300, which is by all accounts a decent camera. If I could afford, say a D3S (I wish), the noise at ISO 3200 would be significantly reduced; but at three times the price of mine, I'll leave that one to the pros.