Noise is a constant problem in digital imaging, although modern cameras are getting better and better at handling noises in the images they capture. Noise in digital photography comes from two main sources.
The first is high ISO noise. This occurs when you turn up a camera's ISO and the increased sensitivity results in a noisy image. Cameras are getting better and better at this and it is becoming less of an issue, however, it is still a problem at extreme sensitivities.
The other area where noise occurs is when you push the exposure of an image and you recover details from the shadow areas. Some cameras are better than others, but because of the way digital imaging works, there is less data in the lower half of an image's brightness values than the top half (this is why people often suggest shooting to the right to optimize the quality of digital images) and when you recover details from the shadows you can often encounter noise. The level of noise varies from camera to camera...