ISO-speeds: What is the difference?

Many new photographers ask the question: "What is ISO-speed and how does it affect my photos?".
I didn't know this when I started shooting with my DSLR, I did a little searching and figured it out:
In a film-camera, the ISO-speed determines the light-sensitivity of the film surface, in a DSLR it determines the light-sensitivity of the sensor.

So what does this mean to you, when you are taking pictures?
Basically it is very essential to getting good pictures at different light levels, choosing the correct ISO-speed can mean the difference between getting a clear shot or a grainy/blurry/underexposed one.
Many people wrongfully believe that ISO1600 is the best thing to take pictures at, giving the best photo, why?

 - It gives shorter shutter times.
 - It takes up more space on the memory card.
 - It drains the camera battery more.
 These are all things that would make a camera beginner think: "It gives all those disadvantages, so the image quality must be better for it!".

But in fact there are entirely different reasons for all of this:
The shorter shutter time, is due to the increased light-sensitivity making the camera able to create an image with less exposure time.
The space it takes up is due to noise, noise takes up more space, than actual image data.
The sensor uses more power when it's sensitivity is increased.
The actual result of changing the ISO-speed can be seen in the example below:

All taken with a Canon EOS 1000D with a Canon EF-S 18-55mm MKII, taken at Av with f/22 (enlarged to show quality)

When the image is not enlarged, the grainy, high ISO images looks the same as the others.

As it can clearly be seen, the ISO100 to ISO400 are pretty clear, but anything higher than that, gives a lot of noise.
the camera settings for the photos are as follows (all taken at f/22, because higher exposure times gives more noise, and makes comparing them easier):

ISO100: 3.2seconds shutter time, 2.00mb space required.
ISO200: 1.6seconds shutter time, 2.23mb space required.
ISO400: 0.8seconds shutter time, 2.69mb space required.
ISO800: 0.5seconds shutter time, 3.09mb space required.
ISO1600: 0.2seconds shutter time, 3,89mb space required.

So what speed is best?
Under these circumstances: ISO100-ISO400 are pretty much the same quality and at 3.2seconds shutter time, camera shake is a major problem if the camera is handheld, so ISO200 or ISO400 would be the best, if you have a high capacity memory card, go for ISO400 in this case.
In this case low ISO-speeds are best, but if it is dark, you don't wanna use the flash and you don't have a tripod, a high ISO-speed is the way to go.

If you are in doubt and you're in a generic situation, you can't really go wrong with ISO200 or ISO400, if the picture is shaken go higher, if the picture is grainy go lower.

If your lens has Image Stabiliser, you might be able to go to a lower ISO-speed, without the camera shake.
If you are new to manual ISO-speeds and you are shooting something that you will not get a second chance on, play it safe an choose automatic ISO-speed until you have mastered the manual ISO-settings.

Disclaimer:
I am not a professional photographer, all the information in this guide is based on personal experiences, Internet searching and the example images displayed above.
As with any photography-technique trial-and-error IS the way to go, this is just a guide to point you in the right direction.
High ISO-speeds are not bad, they are there for a reason, but don't use them if you can get a good photo at a low speed.