Can someone please explain to me what ISO means in the camera settings.

In an answer to a question about taking pictures at a dance party, I was told to use ISO 1600. What does that number really mean? When should I play with ISO settings, only on night pictures, or also during the day?

I see that its the sensitivy of the sensor. What I dont understand is: why the higher ISO, the more noise?, I am confused; if the sensor is more sensitive shouldn't it get more color or more quality?