ISO in a nutshell is how sensitive your sensor is. In days gone by it was a measure of how sensitive your film was. ISO 100 was for shooting in bright light and ISO 800 or 1600 was for shooting in darker conditions. Point-and-shoot cameras often don't let you play with this and the camera chooses for you but it is an integral part of DSLR settings.
The main purpose for increasing ISO is usually to do with camera-shake. If conditions are getting darker then increasing the ISO makes the film (or sensor) more sensitive and therefore it picks up more light at a faster rate and therefore the shutter speed can be reduced - which helps prevent camera-shake.
Sounds great? Well yes, but there is a trade off as always and that's "noise".
Noise is something people don't really notice until it's pointed out to them... certainly, I didn't. After reading this go and have a look at some pics you've taken in low light conditions and you'll see it when you zoom in. In some cases it can be quite obvious.
Here is an example. These pictures were taken in a restaurant under relatively low light without a flash. The first picture was taken at ISO 200 with f3.5 and a resultant shutter speed of 1/10th of a second. The picture below that was shot at ISO 3200 and f3.5 and allowed a shutter speed of 1/250th of a second.
Obviously the shutter speed of the second picture in much faster and so it's less likely to show camera shake as the chances of you moving the camera much in 1/250th of a second is highly unlikely. However if you compare the 2 close-ups of each picture you will see a grainy artifact on the ISO 3200 picture. That is noise.
Personally, I try to shoot in the lowest ISO possible. Normally, I have the camera locked at ISO 100 or 200 but I never let it go above 800 as the noise on the D90 is quite noticeable above this. There is software to remove noise but the results vary. Therefore my advice would be use as low an ISO as you can and add a flashgun if need be.
Or carry a tripod/monopod with you everywhere to prevent camera shake!!!