The ISO setting in your camera effects how sensitive the sensor is to light. There is a lot of technical math that is well explained for those wanting the absolute answer to what it is on Wikipedia, but this article will focus more on choosing the right ISO setting for your specific shot. The right ISO can make your photo, the wrong ISO can ruin it.
ISO on most cameras starts at 100, some cameras start at 50, but the lower the number, the less sensitive to light, the higher the number, the more sensitive. While that seems pretty straight forward, there is a price to pay as you go higher in the ISO settings, digital noise. High ISO film had grain that was visible, somewhat desirable for photographers. On a whole, digital noise, the equivalent to grain for digital photographers is not a welcomed addition to photographs.
Unlike most settings in your camera, like shutter speed and aperture, changing the ISO probably won't be totally noticeable when you review images in the LCD. The digital noise will show up when you view and edit your photos on a computer though. Ideally you want the lowest ISO setting possible for the least amount of noise, but in any given lighting situation that's not always an option.
Sunny days, outside, 100ISO will be wonderful, however if you are noticing a lot of detail in the shadow area of your photos with your naked eye that you aren't seeing when you review files on the LCD on the back of the camera, bump the ISO up to 200 or even 400.
Indoor photography can be tricky, artificial light combined with natural light coming through the windows can result in constant monitoring and adjusting the ISO level. What looks great at 400ISO in one room might require 800ISO in the next room.
If you're utilizing a tripod, use the lowest ISO setting possible. You can get away with a longer shutter speed because the tripod will stop all motion blur of the camera.
Using a flash can help keep the ISO low and your photos noise-free in a lot of situations. Likewise, if depth of field isn't a concern for you, opening up the aperture will allow more light in, giving a lower ISO option.
When shooting at particularly high ISO settings be mindful of proper exposure. Post-production of high ISO shots that weren't properly exposed to the best of your ability really start to fall apart. Particularly when adjusting the exposure or contrast levels, digital noise is enhanced, often resulting in an unpleasing image.
Grain isn't always bad though! One of the most amazing aspects of film noir styled photographs is the grain in them. Low key black and white images with lots of grain can be a good thing, if they are exposed well and the subject matter holds up to it.
The key t0 choosing the correct ISO is determining what factors need to be emphasized properly. Little to no digital noise is what most people strive for, so shooting at the lowest ISO is usually the best bet, unless a film noir style is more for your liking. Since it's impossible for any guide to give you exacting settings, I suggest everyone bracket shoots ISO in a number of situations and then compares the photos in post production so they can best see what happens with your particular camera. ISO is one of the few things you can't correct because you shoot in RAW mode too. If you accidentally shoot some stuff outside on a nice sunny summer day at 3200ISO and were planning on making life-size poster prints, they will have digital noise in them.